EARTH 

FENCES 
IBRARY 


B     ,?KELEY 

LIBRARY 

UNIVERSITY  OF 
CALIFORNIA 


LIBRARY 


ELEMENTARY  CRYSTALLOGRAPHY 


Published    by  the 

McGraw-Hill    Boot  Company 


»Succe>s.sons  to  tneBooK-Departments  or  the 


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Electrical  World  The  Engineering"  and  Mining  Journal 

The  Engineering  Record  R>wer  and  The  Engineer 

Electric  Railway  Journal  American   Machinist 


ELEMENTARY 

CRYSTALLOGRAPHY 


BEING    PART    ONE    OF 


GENERAL  MINERALOGY 


BY 

W.    S.    BAYLEY,    Ph.  D. 

ASSOCIATE    PROFESSOR   OF   MINERALOGY   AND   ECONOMIC    GEOLOGY 
UNIVERSITY   OF    ILLINOIS. 


McGRAW-HILL  BOOK  COMPANY 

239  WEST  39TH  STREET,  NEW  YORK 

6  BOUVERIE  STREET,  LONDON,  E.  C. 

1910 


COPYRIGHT,  1910 

BY  THE 
McGRAW-HiLL  BOOK  COMPANY 


Printed  and  Electrotyped  by 

The  Maple.  Press 

York,  Pa. 


PREFACE. 


The  material  in  the  following  pages  was  originally  intended 
to  be  the  introductory  portion  of  a  text-book  on  Mineralogy. 
Because  of  the  increasing  interest  in  Crystallography  on  the 
part  of  chemists  and  others,  however,  it  has  been  finally  decided 
to  offer  it  as  an  independent  volume,  in  the  hope  that  it  might 
find  a  wider  use  in  this  form  than  if  it  were  merely  a  portion  of 
a  larger  volume  which  would  appeal  mainly  to  mineralogists. 
It  is  intended  that  it  shall  be  followed  in  the  near  future  by  a 
second  volume  which  will  deal  with  minerals — the  two  taken 
together  being  expected  to  constitute  an  elementary  text-book 
on  Mineralogy. 

The  present  volume  does  not  pretend  to  be  a  treatise  on 
Crystallography.  It  is  merely  a  guide  for  those  attempting  to 
gain  some  insight  into  the  fundamental  principles  underlying 
the  science.  It  is  the  direct  result  of  the  need  felt  by  the  author 
in  his  own  classes  for  a  simple  discussion  of  crystals  and  a  series 
of  simple  statements  of  crystallographic  conceptions.  It  is  not 
intended  to  be  a  description  of  crystal  forms,  nor  an  illustrated 
list  of  crystallographic  symbols.  It  is  hoped  that  it  may  serve 
as  an  aid  in  teaching,  but  is  not  expected  to  serve  as  a  reference 
book.  Consequently  some  of  the  facts  usually  contained  in 
text-books  on  Crystallography  will  not  be  found  in  its  pages. 
Whether  the  material  retained  has  always  been  chosen  with 
wisdom  must  be  left  to  the  judgment  of  its  readers.  It  was 
written  for  students  of  Mineralogy,  Chemistry,  and  Physics. 
The  mineralogist  has  always  used  crystallography  as  an  effi- 
cient aid  in  his  study  of  minerals.  The  chemist  is  beginning 
to  appreciate  its  value  in  his  work.  The  physicist  will  before 
long  discover  that  it  is  absolutely  necessary  to  employ  its  methods 


819467 


VI  PREFACE 

in  studying  the  physical  properties  of  many  of  the  substances 
with  which  he  has  to  deal  before  he  may  hope  to  reach  com- 
parable results. 

Because  of  the  use  to  which  the  book  is  to  be  put,  the  author 
has  assumed  but  little  preliminary  knowledge  of  mathematics, 
physics,  and  chemistry  on  the  part  of  its  users.  Technical  terms 
have  been  employed  as  rarely  as  possible,  consistent  with  clear- 
ness, and  only  those  topics  are  discussed  which  are  necessary  to 
the  understanding  of  the  nature  of  crystals.  The  discussion 
of  optical  phenomena  has  been  reduced  to  a  minimum,  because 
of  their  complex  nature,  and  the  difficulty  of  observing  them 
without  a  supply  of  expensive  apparatus. 

The  choice  of  the  Naumann  System  of  parameter  symbols 
in  preference  to  the  more  elegant  Miller  System  of  indices  has 
been  made  with  deliberateness,  solely  because  of  the  greater 
ease  with  which  the  Naumann  symbols  are  comprehended  by 
students  who  approach  the  study  of  crystals  for  the  first  time. 
The  indices,  however,  are  indicated  for  all  the  forms  described 
or  pictured. 

There  is  no  claim  that  any  portion  of  the  volume  is  original, 
except  perhaps  its  method  of  developing  the  discussion.  The 
illustrations  have  been  collected  from  various  sources.  A  few 
are  original.  Others  have  been  taken  from  the  text-books  of 
Williams,  Groth  arid  Linck.  When  possible  to  do  so,  their 
sources  have  been  indicated. 

The  writer  is  under  obligations  to  Dr.  R.  M.  Bagg,  Jr.,  for 
photographs  of  some  of  the  minerals  illustrated  in  the  text  and 
to  the  publishers  for  their  painstaking  efforts  to  give  the  volume 
an  attractive  dress. 


CONTENTS. 


Introduction.  Mineralogy:  Its  Object,  History,  and  Division 

into  Branches    . 


IX 


CHAPTER 
CHAPTER 

I. 

II. 

CHAPTER 
CHAPTER 
CHAPTER 

III. 
IV. 
V. 

CHAPTER 

VI. 

CHAPTER 

VII. 

CHAPTER  VIII, 


CHAPTER 


CHAPTER 


IX. 


X. 


CHAPTER  XL 

CHAPTER  XII. 

CHAPTER  XIII. 

CHAPTER  XIV. 

CHAPTER  XV. 


PART  I. 

GEOMETRICAL  CRYSTALLOGRAPHY. 

General  Facts  of  Crystallography      .....  3 
The    Law    of    the    Constancy    of    Interfacial 

Angles 14 

The  Law  of  Simple  Mathematical  Ratios   .    .  17 

Symmetry      25 

The    Isometric,   or    Regular,    System — Holo- 
hedral Division 31 

Partial  Forms — Hemihedrism  and  Tetartohe- 

drism  of  the  Isometric  System 41 

The  Hexagonal  System 54 

Holohedral  Division 60 

Hemihedral  Division 69 

Tetartohedral  Division 82 

The  Tetragonal  System 87 

Holohedral  Division 87 

Hemihedral  Division 95 

The  Orthorhombic  System 102 

Holohedral  Division 105 

Hemihedral  Division   . 112 

The  Monoclinic  System      115 

Holohedral  Division    . 117 

Hemihedral  Division 121 

The  Triclinic  System 125 

Crystal  Imperfections 132 

Crystal  Aggregates 143 

Amorphous  Substances  and  Pseudomorphs   .  160 

Crystal  Projection .164 

vii 


VI 11 


CONTENTS. 

PART  II. 


PHYSICAL  CRYSTALLOGRAPHY. 
CHAPTER     XVI.     Introduction:     Physical  Agencies    and   Phys- 


CHAPTER  XVII. 
CHAPTER  XVIII. 
CHAPTER  XIX. 
CHAPTER  XX. 


ical  Symmetry '    .  177 

Mechanical  Properties  of  Crystals 180 

Optical  Properties  of  Crystals    .    .    .    .    .    .    .196 

Thermal  Properties  of  Crystals 209 

Electrical  and  Magnetic  Properties  of  Crystals .  214 

PART  III. 

CHEMICAL  CRYSTALLOGRAPHY. 


CHAPTER     XXI. 
CHAPTER  XXII. 


Solution  and  Etched  Figures.    . 
Isomorphism  and  Polymorphism 


219 

227 


INTRODUCTION. 


MINERALOGY:     ITS   OBJECT,    HISTORY,   AND 
DIVISIONS  INTO  BRANCHES. 

The  Purpose  of  Mineralogy  is  the  study  of  the  solid,  homo- 
geneous, inorganic  compounds  occurring  in  nature,  either  as  the 
constituents  of  rock  masses  or  as  the  components  of  veins,  which 
are  the  fillings  of  chinks  or  fissures  in  the  earth's  crust. 

The  science  is  closely  related  to  Chemistry,  since  the  objects 
of  its  study  are  definite  chemical  compounds,  all  of  whose  proper- 
ties are  believed  to  be  determined  by  their  composition,  and  to 
Geology,  because  their  aggregations  make  up  by  far  the  greater 
portion  of  the  earth's  crust— the  rocks. 

The  principles  made  use  of  in  the  study  of  minerals  are  not 
peculiar  to  the  science  of  Mineralogy.  Since  minerals  are 
definite  chemical  compounds  they  must  be  studied  in  part  by 
chemical  methods.  The  properties  by  which  they  are  known 
are  partly  geometrical  and  partly  physical,  and  these  must  be 
investigated  by  mathematical  and  physical  processes.  Miner- 
alogy is  thus  not  a  pure  science  with  methods  of  study  peculiar 
to  itself.  It  is  a  mixed  science.  Its  problems  are  attacked  by 
methods  that  are  borrowed  from  other  sciences  and  applied  to 
minerals  for  the  purpose  of  discovering  their  nature. 

Distinction  between  Minerals  and  Rocks. — A  mineral 
is  any  definite  inorganic  compound,  either  solid  or  liquid,  occur- 
ring in  nature.  It  is  not  the  direct  result  of  life  processes  nor 
the  product  of  man's  experiments. 

A  rock  is  a  definite  portion  of  the  earth's  crust  irrespective  of 
its  composition.  It  is  usually  a  mass  of,  sufficient  size  to  be 
regarded  as  an  architectural  unit  in  the  structure  known  as  the 
earth's  crust.  It  does  not  necessarily  possess  a  definite  chemical 
composition,  nor  is  it  usually  homogeneous.  It  may  be  an  aggre- 

ix 


X  INTRODUCTION 

gate  of  minerals  of  the  same  kind,  as  limestone,  which  is  composed 
of  grains  of  the  mineral  calcite  (CaCO3),  or  it  may  be  a  mixture 
of  minerals  of  different  kinds,  as  granite,  which  consists  es- 
sentially of  feldspar,  (a  potassium-aluminium  silicate),  and  quartz 
(SiO2),  or  it  may  be  an  accumulation  of  organic  substances,  like 
coal,  or  a  mass  of  undifferentiated  glass. 

Different  portions  of  the  same  rock  when  analyzed  will 
usually  give  different  results,  and  in  only  the  few  rare  instances 
where  a  single  mineral  may  constitute  its  entire  mass  can  its 
composition  be  represented  by  a  simple  chemical  formula. 

According  to  the  definitions  given,  water  (H2O)  is  a  mineral, 
as  are  also  quartz  (SiO2)  magnetic  iron  ore  (Fe3O4),  diamond 
(C),  etc.  Coal,  amber,  and  wood  are  not  minerals,  because  they 
are  the  products  of  organic  processes.  Granite  and  obsidian 
(volcanic  glass)  are  not  minerals  because  they  do  not  possess 
definite  chemical  compositions.  Tin  is  not  a  mineral  because 
it  is  the  product  of  man's  work,  while  natural  tin  oxide,  the  ore 
from  which  the  metal  is  obtained,  is  the  mineral  cassiterite  or 
tin  stone.  The  copper  that  occurs  as  native  copper  is  a  mineral, 
while  the  copper  obtained  by  smelting  the  sulphide  is  an  artificial 
product. 

History  of  Mineralogy.— Although  the  study  of  the  metals 
and  of  gems  dates  back  certainly  to  the  time  of  Aristotle  (384 
B.  C.),  the  scientific  study  of  the  less  valuable  minerals  began 
with  Haiiy  (1743),  a  famous  French  teacher,  who  discovered 
that  all  minerals  possess  characteristic  forms  which  may  be 
deduced  in  some  cases  from  their  physical  properties.  In  the 
last  half  of  the  eighteenth  century  and  in  the  early  years  of  the 
nineteenth  century,  Werner  (1750)  and  Weiss  (1808),  in  Germany, 
described  in  great  detail  a  large  number  of  minerals,  and  the  latter 
author  developed  a  method  of  classifying  and  representing  their 
forms.  From  this  time  until  the  present  day  Germans  have  been 
the  most  earnest  students  of  Mineralogy  in  all  its  branches. 
France,  Great  Britain,  and  Italy  have  many  devoted  mineralogists, 
while  North  America  has  produced  quite  a  number  who  have 
acquired  world-wide  reputations. 

The  science  has  recently  become  so  broad  that  no  one  pretends 


INTRODUCTION  XI 

to  be  versed  in  all  its  branches.  There  are  descriptive  miner- 
alogists, physical  mineralogists,  optical  mineralogists,  mathe- 
matical mineralogists,  and  chemical  mineralogists,  the  members 
of  each  class  devoting  themselves  to  special  phases  of  the  science, 
but  all  being  more  or  less  thoroughly  acquainted  with  the  results 
of  the  work  in  all  its  other  phases. 

Divisions  of  the  Subject. — Since  Mineralogy  is  such  a  broad 
science,  dealing  as  it  does  with  the  geometrical,  physical,  and 
chemical  properties  of  minerals,  it  has  been  found  convenient  to 
divide  it  into  branches,  each  one  of  which  is  devoted  to  the  study 
of  some  special  class  of  properties. 

The  fundamental  basis  of  mineralogy  is  chemistry.  All 
minerals  are  definite  chemical  compounds,  the  properties  of  each 
one  of  which  are  directly  dependent  upon  their  chemical  com- 
position and  structure.  Thus  Chemical  Mineralogy  is  one 
branch  of  the  subject,  and  upon  it  all  the  other  branches  are 
founded. 

When  different  chemical  compounds  are  studied  it  is  soon 
discovered  that  they  vary  in  color,  hardness,  ease  of  fusibility, 
etc.,  i.e.,  in  their  physical  properties.  We  therefore  recognize  a 
branch  of  mineralogy  that  is  devoted  to  the  investigation  of  the 
physical  properties  of  minerals— Physical  Mineralogy. 

Again,  careful  scrutiny  of  these  same  compounds  soon 
reveals  the  fact  that  individual  minerals  are  characterized  by 
definite  and  distinctive  forms  by  which  they  may  be  recognized. 
Thus  a  piece  of  quartz  (SiO2)  may  often  be  distinguished  from  a 
piece  of  calcite  (CaCO3)  simply  by  its  external  form.  Morpho- 
logical Mineralogy  deals  with  the  forms  of  minerals.  But 
distinctive  forms  are  not  confined  to  minerals.  They  are  pos- 
sessed likewise  by  other  substances  which  have  been  produced 
under  conditions  analogous  to  those  under  which  minerals  are 
formed.  They  are  characteristic  of  nearly  all  solid  substances 
that  have  been  sublimed  from  gases  or  vapors  or  have  been 
precipitated  from  solutions.  The  study  of  such  forms  from  the 
general  point^  of  view  is  known  as  Crystallography.  Further, 
bodies  exhibiting  these  definite  geometrical  forms  are  also 
characterized  by  certain  well-defined  physical  properties  which 


Xll  INTRODUCTION 

are  closely  related  to  the  forms  exhibited.     Consequently,  Crystal 
lography  is  naturally   divisible   into   two   branches,   viz.,   Geo- 
metrical Crystallography  and  Physical  Crystallography. 

When  minerals  are  studied  with  respect  to  their  individual 
properties,  we  have  Descriptive  Mineralogy. 

After  becoming  acquainted  with  the  characteristics  of  different 
minerals  it  often  becomes  necessary  to  make  use  of  these  charac- 
teristics in  determining  the  nature  of  an  unknown  substance. 
This  application  of  our  knowledge  concerning  the  properties  of 
known  minerals  to  the  discovery  of  the  nature  of  an  unknown 
mineral  is  called  Determinative  Mineralogy.  In  Chemistry  it 
corresponds  to  Analytical  Chemistry. 

The  present  volume  deals  only  with  the  elements  of  Geo- 
metrical and  Physical  Crystallography.  A  later  volume  will  be 
devoted  to  a  discussion  of  the  general  characteristics  of  minerals 
and  descriptions  of  the  features  of  the  most  common  kinds. 


PART  I. 

GEOMETRICAL  CRYSTALLOGRAPHY. 


CHAPTER  I. 


GENERAL  FACTS  OF  CRYSTALLOGRAPHY. 

Forms  Assumed  by  Substances. — The  forms  assumed  by 
substances  through  the  agency  of  natural  forces  are  of  two  classes : 
(i)  those  produced  by  the  action  of  internal  molecular  forces 
and  (2)  those  produced  by  external  agencies.  The  first  depend 
upon  the  nature  of  the  material  composing  the  substance,  and 
are  known  as  Idiomorphic  forms;  the  second  are  to  a  large  extent 
independent  of  the  nature  of  the  substance,  but  are  determined 
by  the  nature  of  the  mechanical  agents  acting  upon  it  and  the 
directions  along  which  they  act,  or  by  the  methods  by  which  it 
was  produced.  They  are  accidental  forms.  Since  the  forms 


FIG.   i. — Crystals  of  calcite 
attached  at  one  end. 


FIG.  2.— Stalactite  of  calcite. 


of  this  class  are  not  directly  dependent  upon  the  nature 'of  the 
bodies  exhibiting  them,  they  cannot  be  classified  and  studied 
systematically. 

Figure  i  illustrates  an  idiomorphic  form  (a  group  of  crystals 
of  calcite  (CaCO3)),  and  figure  2,  an  accidental  form  (a  stalactite 
of  the  same  substance).  The  shape  of  the  crystals  (Fig.  i)  is  due 
to  the  fact  that  they  are  crystals  of  calcite.  Crystals  of  other  sub- 
stances have  not  the  same  shape  as  this  one.  The  shape  of  the 
stalactite  (Fig.  2)  is  due  to  the  fact  that  the  calcite  of  which  it  is 

3 


4*  GEOMETRICAL    CRYSTALLOGRAPHY 

composed  was  deposited  from  dripping  water  and  not  to  the  fact 
that  its  substance  is  calcite.  There  are  stalactites  of  many  sub- 
stances not  at  all  similar  to  calcite.  Among  other  forms  of  the 
second  class  may  be  mentioned  those  of  mineral  pebbles,  frag- 
ments, cut  stone,  like  the  sets  in  rings,  etc.,  all  of  which  were 
produced  by  forces  which  had  their  origin  outside  the  substance. 

Idiomorphic  Forms. — On  the  contrary,  the  idiomorphic 
forms  are  directly  dependent  upon  the  nature  of  the  substance  of 
which  they  are  characteristic.  Calcite  (CaCO3)  always  possesses 
definite  forms  that  are  distinctive  for  this  mineral ;  that  is,  exactly 
the  same  form  is  never  seen  in  other  minerals.  Quartz  (SiO2) 
also  has  its  distinctive  forms,  which  are  peculiar  to  itself.  So 
with  many  other  substances. 

Occasionally  several  substances  which  are  apparently  iden- 
tical in  chemical  composition  may  have  different  idiomorphic 
forms,  as  is  the  case  with  calcium  carbonate  (CaCO3).  The 
apparent  identity  of  such  substances  is,  however,  believed  not 
to  be  a  real  identity.  There  are  two  calcium  carbonates  which 
differ  from  one  another  in  the  structure  of  their  chemical  molecule 
or  in  the  way  in  which  their  molecules  are  united.  They  are, 
therefore,  different  chemical  compounds.  These  calcium  car- 
bonates possess  different  idiomorphic  forms,  as  we  should  expect 
of  them,  and,  therefore,  they  are  given  different  names.  One  of 
them  is  the  mineral  calcite  and  the  other  the  mineral  aragonite. 

Idiomorphic  forms  may  be  defined  as  those  which  are  original 
to  the  body  on  which  they  occur,  and  which  are  essential  to  its 
substance. 

Molecules  and  Crystal  Particles. — All  bodies  are  believed 
to  be  made  up  of  tiny  particles  called  molecules,  which  vary  in 
size  and  weight  for  different  substances.  They  are  so  very  small 
that  the  most  powerful  microscope  will  not  detect  them.  In  a 
single  cubic  inch  of  a  cold  gas  like  nitrogen  they  are  present  to  the 
number  of  about  no  billion  billions.  These  molecules  are  so 
small  that  it  would  take  about  55,000,000  to  make  a  row  one  inch 
long.  Nevertheless,  if  all  thaT  are  present  in  a  cubic  inch  of  gas 
were  placed  in  a  row,  touching  one  another,  they  would  form  a 
line  32,000,000  miles  long.  The  molecules  are  always  in  motion, 


GENERAL    FACTS    OF    CRYSTALLOGRAPHY  5 

but  they  are  supposed  not  to  touch  each  other  even  in  the  hardest 
and  densest  bodies.  They  are  separated  from  one  another  by  tiny 
spaces. 

When  a  body  passes  from  the  liquid  or  gaseous  state  to  the 
solid  form,  the  little  molecules  tend  to  arrange  themselves  into 
definite  groupings,  which  are  the  crystal  particles.  These  are 
built  up  by  the  addition  of  other  particles  until  they  become  large 
enough  to  be  seen  first  by  the  microscope,  then  by  the  naked  eye, 
until  finally  they  reach  a  size  that  may  be  measured  by  inches  or 
even  feet.  All  the  little  particles  of  a  body  built  u£  in  this  way 
are  constructed  on  the  same  plan,  so  that  every  portion  of  the 
body  has  the  same  structure.  If  we  should  take  bits  from  its 
different  parts  and  could  magnify  them  sufficiently  to  render  their 
component  molecules  visible,  we  would  find  these  little  bodies  all 
arranged  in  a  definite  manner,  which  would  be  the  same  for  every 
fragment.  If  we  could  examine  other  bodies  in  the  same  way, 
we  should  find  that  they,  too,  are  composed  of  molecules 
arranged  in  certain  definite  groupings;  but,  in  many  cases, 
we  should  discover  these  groupings  to  be  different  from  those  of 
the  first  body  studied. 

While  we  cannot  see  the  method  of  grouping  of  the  particles, 
we  can  infer  something  of  its  character  from  the  phenomena 
presented  by  different  kinds  of  bodies.  From  the  effects  produced 
we  can  study  the  groupings  themselves,  and  thus  can  gain  an 
insight  into  the  internal  structure  of  the  bodies  which  they 
compose. 

From  a  careful  consideration  of  the  subject  from  all  points  of 
view  it  is  concluded  that  the  arrangement  of  the  molecules  in  any 
solidifying  body  is  dependent  almost  exclusively  upon  the  nature 
of  the  little  molecules  themselves.  It  is  believed  that  the  power 
to  build  up  well-defined  and  characteristic  groupings  is  a  property 
of  the  molecules,  just  as  the  power  of  chemical  combination 
between  the  atoms  is  an  inherent  property  of  the  atoms.  But 
the  chemical  composition  of  every  homogeneous  body  is  the  same 
as  the  chemical  composition  of  its  component  molecules,  hence 
we  may  infer  that  the  form  assumed  by  a  body  is  dependent 
upon  its  composition. 


6  GEOMETRICAL   CRYSTALLOGRAPHY 

In  order  to  explain  the  peculiarities  of  a  body  in  its  different 
states  it  seems  necessary  to  assume  that  every  molecule  possesses 
attractive  powers  by  which  it  tends  to  pull  toward  itself  other 
molecules  of  the  same  kind,  and  that  at  the  same  time  it  is  endowed 
with  the  impulse  to  move  in  straight  lines.  These  two  properties 
which  are  supposed  to  be  inherent  in  every  molecule  tend  to 
counteract  one  another  in  part.  When  the  tendency  to  move 
away  from  its  neighbors  (the  repulsive  tendency)  predominates 
in  the  molecules  composing  a  body,  that  body  is  said  to  be  a  gas. 
When  the  attractive  tendency  overcomes  the  repulsive  one,  the 
body  is  a  liquid  or  a  solid.  The  distinction  between  the  solid 
and  the  liquid  states  depends  upon  other  considerations — such, 
perhaps,  as  the  friction  between  their  constituent  molecules. 
The  two  pass  gradually  into  one  another  so  that  it  is  often  diffi- 
cult to  discriminate  between  them.  A  body  is  usually  said  to  be 
a  liquid  if  it  assumes  a  spherical  form  when  floating  in  a  medium 
with  which  it  does  not  mix. 

Crystalline  Bodies  and  Crystals. — Not  all  solid  substances 
are  constructed  of  regularly  arranged  molecules.  In  some  the 
molecules  possess  no  regularity  of  arrangement,  so  far  as  we  can 
learn.  This  class  includes  such  bodies  as  glasses,  jellies,  etc., 
which  are  spoken  of  as  amorphous  or  colloidal.  Other  substances, 
like  the  tissues  of  plants  and  animals,  are  constructed  of  cells 
regularly  built  up,  but  this  regularity  is  not  the  result  of  the  piling 
up  of  the  molecules  in  a  definite  manner  in  consequence  of  any 
property  inherent  in  them,  but  is  the  result  of  the  life  processes 
of  the  animals  or  plants  that  make  the  cells.  The  molecules 
in  the  cells  are  not  regularly  and  definitely  arranged,  although 
the  shapes  of  the  little  cells  themselves  may  all  be  alike.  In 
other  words,  while  the  tissue  is  composed  of  regularly  arranged 
cells,  the  materials  composing  the  cells  may  be  amorphous. 

When  a  solid  or  fluid  body  is  composed  throughout  of  molecules 
arranged  according  to  some  definite  plan,  it  is  said  to  be  crystalline; 
when  composed  of  molecules  arranged  helter-skelter,  i.e.,  without 
any  definite  plan,  it  is  said  to  be  amorphous  or  colloidal. 

The  distinction  between  an  amorphous  body  of  definite  form 
and  a  crystalline  body  may  be  easily  shown  with  the  aid  of  a  few 


GENERAL   FACTS    OF   CRYSTALLOGRAPHY  7 

buckshot.  If  a  layer  of  these  be  built  into  a  triangle,  as  illustrated 
in  figure  3  a,  and  on  this  layer  a  second  layer  be  placed  in  the 
manner  indicated,  and  upon  this  a  third  layer,  and  so  on,  we 
finally  obtain  a  pile  of  shot  constructed  of  regularly  arranged 
components  (Fig.  3  b).  The  shape  of  the  pile  depends  upon  the 
plan  of  the  foundation  triangle  and  upon  the  manner  in  which 
the  successive  layers  are  piled  upon  the  first  one.  If  we  let  the 
shot  represent  molecules,  the  pile  represents  a  crystalline  body 
with  a  regular  internal  structure.  On  the  other  hand,  we  may 
tumble  the  shot  indiscriminately  into  several  little  boxes  all  made 
exactly  alike.  The  masses  of  shot  within  the  boxes  would  all 


have  the  same  shapes — the  shape  of  the  boxes — but  there  would 
be  no  regularity  in  the  arrangement  of  the  shot.  The  contents 
of  the  boxes  illustrate  the  structure  of  an  amorphous  body. 

Since  crystalline  bodies  exhibit  their  physical  properties  in 
different  degrees  along  different  directions,  a  second  assumption 
with  regard  to  molecules  is  necessary,  viz.,  that  their  activities  are 
oriented.  In  other  words,  the  lines  along  which  their  activities 
are  exerted  are  definitely  arranged  with  respect  to  one  another. 
When  we  are  dealing  with  an  amorphous  body  in  which  there  is 
no  regularity  in  the  arrangement  of  its  molecules,  there  is,  as  a  con- 
sequence, no  regularity  in  the  arrangement  of  the  directions  along 
which  their  powers  act;  consequently  such  bodies  possess  similar 
properties  in  parallel  directions.  Thus  a  sphere  of  glass,  which 


8  GEOMETRICAL    CRYSTALLOGRAPHY 

is  amorphous,  remains  a  sphere  under  all  conditions  of  tempera- 
ture, i.e.,  it  expands  equally  in  all  directions.  On  the  other  hand, 
a  sphere  of  quartz,  which  is  crystalline,  loses  its  spherical  form 
when  the  temperature  rises  or  falls  and  becomes  ellipsoidal,  i.e., 
it  expands  and  contracts  to  a  greater  degree  in  one  direction 
than  in  others.  Its  physical  properties  exhibit  a  definite 
arrangement. 

Crystals. — When  the  little  molecules  of  crystalline  bodies  are 
given  the  opportunity  and  the  time  to  arrange  themselves  in 
accordance  with  the  plan  that  suits  them  best,  they  build  up 
structures  usually  bounded  by  planes,  which  vary  in  number  and 
in  inclination  to  one  another  according  to  certain  definite  laws. 
A  homogeneous  crystalline  body  thus  bounded  by  planes  is  called 
a  crystal,  or  a  crystal  individual.  Or,  since  the  bounding  planes 
are  the  result  of  the  internal  structure,  a  crystal  may  be  defined  as  a 
homogeneous  body  bounded  by  a  polyhedron  that  is  idiomorphic. 

The  term  homogeneous  is  introduced  into  the  definition 
because  it  sometimes  happens  that  a  single  body  bounded  by 
plane  faces  consists  of  several  parts  united  in  such  a  way  that, 
while  the  composition  and  arrangement  of  the  molecules  in  all 
portions  of  each  part  is  the  same,  the  composition  and  arrange- 
ment in  the  different  parts  is  different.  The  term  homogeneous 
signifies  that  all  parts  of  the  individual  are  like  all  other  parts. 

Although  the  great  majority  of  crystalline  bodies  are  solid, 
there  are  some  liquids  which  under  proper  conditions  exhibit  a 
definite  internal  structure  which  is  recognizable  in  polarized  light 
(see  page  196)  and  which,  when  in  the  form  of  drops,  show  a 
tendency  to  bound  themselves  by  plane  surfaces.  These  are 
known  as  liquid  crystals.  They  do  not  possess  the  sharply  defined 
forms  of  solid  crystals,  but  are  distinctly  bounded  by  surfaces  that 
are  nearly  plane  and  that  intersect  in  edges  (see  Fig.  4). 

Crystallization. — The  process  by  which  crystals  and  crystal- 
line bodies  are  formed  is  known  as  crystallization.  No  satis- 
factory explanation  of  the  force  of  crystallization  has  yet  been 
proposed.  It  is  known  that  some  substances  possess  this  power 
to  a  much  greater  degree  than  others,  and  in  some  (the  colloids) 
it  appears  to  be  entirely  lacking.  In  the  latter  cases,  however, 


GENERAL    FACTS    OF    CRYSTALLOGRAPHY  9 

the   power  may  still  exist,  but  if  so  it  is  so  weak  that  it  can 
express  itself  only  under  the  most  favorable  conditions. 

Crystallography. — The  external  forms  of  crystallized  bodies 
are  their  most  striking  peculiarities,  consequently  they  have  been 
studied  longer  than  their  other  features.  They  were  at  first 
supposed  to  be  their  most  essential  characteristics,  and  hence 
their  study  was  early  named  crystallography.  It  is,  however, 
now  well  recognized  that  the  forms  are  only  one  of  the  methods 
through  which  the  molecular  structure  of  a  crystallized  body 
expresses  itself.  The  term  as  now  used  includes  not  only  the 


FIG.  4. — Photograph  of  liquid  crystals  with  rounded  edges. 
(After  O.  Lehmann.) 


study  of  the  forms  of  crystals,  but  also  the  study  of  all  their 
other  characteristic  properties.  Sometimes  the  former  is  dis- 
tinguished as  geometrical  crystallography  and  the  latter  as  physical 
and  chemical  crystallography.  Crystallography  includes  the 
study  of  the  forms  assumed  by  all  crystallized  bodies,  whether 
these  bodies  occur  as  minerals  or  whether  they  exist  only  as  the 
products  of  the  laboratory.  When  applied  to  mineral  substances 
it  is  often  called  ''Morphological  Mineralogy." 

Laws  of  Crystallography. — Almost  as  soon  as  the  forms  of 
crystals  began  to  be  studied  carefully  it  was  discovered  that  the 
planes  which  comprise  them  are  fixed  in  position  and  that  their 
relations  to  one  another  are  so  well-defined  that  they  may  be  ex- 
pressed mathematically.  The  crystallographic  laws  thus  dis- 
covered are  very  simple  and  are  the  foundation  of  all  modern 
crystallography.  They  are  three  in  number,  as  follows : 

i.  The  law  of  the  constancy  of  the  interfacial  angles  on  all 
crystals  of  the  same  substance. 


10 


GEOMETRICAL   CRYSTALLOGRAPHY 


2.  The  law  of  the  rationality  of  the  indices;  or  the  law  of  sim- 
ple mathematical  ratios. 

3.  The  law  of  symmetry. 

Definitions. — Before  attempting  to  discuss  the  meanings  of 
the  laws  and  their  significance  it  is  necessary  to  become  acquainted 
with  some  of  the  commonest  terms  used  in  describing  crys- 
tals. These  are  not  many  in  number  nor  are  they  difficult  of 
comprehension. 

An  interfacial  angle  is  the  angle  included  between  any  two  of 
the  planes  or  faces  of  a  crystal  or  between  their  prolongations 
(ABC  in  Fig.  5). 


FIG.  6. — Zo  al  arrangement  of 
planes  on  crystal. 


A  crystal  angle  is  the  solid  angle  in  which  three  or  more  faces 
meet  (D,  E  in  Fig.  5). 

A  crystal  edge  or  interfacial  edge  is  the  line  in  which  two  con- 
tiguDus  crystal  faces  meet  (E-E,  D-D  in  Fig.  5). 

A  zone  of  planes  is  a  belt  of  planes  whose  edges  with  each  other 
are  parallel  lines  (see  Fig.  6). 

A  zonal  axis  is  a  line  passing  through  the  center  of  a  crystal 
parallel  to  the  edges  of  a  zone  of  planes.  All  the  planes  lying  in  a 
zone  are  parallel  to  the  zonal  axis  for  that  zone  (see  Fig.  6) .  The 
planes  c,o,o2,m,o2,o  in  front,  constitute  one  zone  on  the  crystal 
illustrated  in  figure  6,  o,q,s,m,s,q  form  another,  and  the  planes 


GENERAL   FACTS    OF    CRYSTALLOGRAPHY 


II 


marked  m  a  third.  The  zonal  axis  of  the  third  zone  is  the  line 
AB,  that  of  the  first  zone  is  CD,  and  that  of  the  second  zone  EF, 
provided  the  lines  are  regarded  as  passing  through  the  center  of 
the  crystal.* 

Measurement  of  Interfacial  Angles. — In  order  to  determine 
the  elements  of  crystals  it  is  necessary  that  the  vajfcies  of  their 
interfacial  angles  should  be  known,  as  the  inclination  of  the  plajie? 
to  one  another  varies  with  the  values  of  these  angles,  or,  to  put 


FIG.  7. — Contact  goniometer. 


FIG.  8. — Simple  form  of  con- 
tact goniometer. 


the  case  more  logically,  the  values  of  the  interfacial  angles  vary 
with  the  inclination  of  the  planes  including  them. 

The  instruments  with  which  the  interfacial  angles  are  meas- 
ured are  known  as  goniometers.  The  simplest  form  is  the 
contact  goniometer  (Fig.  7).  This  form  consists  of  a  graduated 
arc  and  two  detachable  arms,  one  of  which  revolves  about 
a  pivot  common  to  both.  These  arms  are  removed  from  the 
arc  and  applied  to  the  two  faces  of  the  crystal  whose  angles  are 
to  be  measured,  care  being  taken  to  hold  them  perpendicular 
to  the  interfacial  edge,  and  at  the  same  time  to  press  them  firmly 

*  Before  entering  further  upon  the  study  of  crystals  the  student  should  familiar- 
ize himself  with  the  terms  denned,  by  applying  them  to  the  proper  parts  of  a  few 
simple  crystals  or  crystal  models.  First  the  edges,  the  interfacial  and  the  crystal 
angles  should  be  pointed  out,  and  then  all  the  zones  occurring  on  a  few  more  com- 
plicated models  should  be  determined.  A  very  useful  and  cheap  set  of  60  crystal 
models  in  wood  may  be  obtained  from  dealers  in  minerals.  Other  and  more 
comprehensive  sets  may  be  obtained  from  Dr.  F.  Krantz,  Bonn  on  Rhine, 
Germany,  at  an  average  cost  of  45  cents  for  each  model. 


12 


GEOMETRICAL   CRYSTALLOGRAPHY 


against  the  faces.  The  screw  holding  the  arms  is  then  set,  and 
the  arms  themselves  are  replaced  on  the  arc,  when  the  value  of  the 
angle  desired  is  indicated  by  the  graduation.  Measurements 
made  with  the  contact  goniometer  are  reliable  to  about  half  a 
degree. 

Other  forms  of  the  contact  goniometer  are  even  simpler  than  the 
one  described.  Their  construction,  however,  is  so  similar  to  that 
of  the  one  illustrated  that  they  demand  no  description  (see  Fig.  8). 

A  more  accurate  but  much  more  complicated  instrument  is  the 
reflection  goniometer  (Fig.  9).  Its  use  depends  upon  the  prin- 


FIG.  9. — Reflection  goniometer. 

ciple  that  a  ray  of  light  falling  upon  two  planes  in  parallel  posi- 
tion is  reflected  from  them  in  parallel  directions.  The  mechan- 
ism of  the  instrument  is  such  as  to  give  a  fixed  ray  of 
light  (through  L  in  the  figure),  a  means  of  detecting  its  reflection 
(O),  and  an  arrangement  for  so  placing  a  crystal  that  its  con- 
tiguous planes  may  be  brought  successively  into  a  given  position 
(K).  By  the  use  of  a  well-made  instrument  the  angle  through 
which  the  crystal  must  be  turned  in  order  that  its  faces  shall  give 


GENERAL    FACTS    OF    CRYSTALLOGRAPHY  13 

parallel  reflections  can  be  read  off  to  within  a  fraction  of  a 
minute  of  arc  on  the  graduated  circle  V.  This  angle  is  the 
supplement  of  the  interfacial  angle. 

Let  AOC  (in  Fig.  10)  be  the  interfacial  angle  to  be  measured 
and  EP  a  ray  of  light  falling  upon  the  face  AO.  It  will  be  re- 
flected along  the  line  PG.  In  order  that  the  same  ray  shall  be 
reflected  from  the  face  CO  in  the  same  direction  this  face  must 
be  brought  into  the  position  OC',  when  OA  will  have  the  position 
OA'.  But  in  order  to  bring  OC  into  the  position  OC'  the 
crystal  must  be  turned  through 
the  angle  COC',  which  is  the 
supplement  of  the  interfacial 
angle  COA. 

Importance  of  Accurate 
Measurements  of  Interfacial 
Angles. — The  importance  of  de- 
termining accurately  the  values 
of  interfacial  angles  is  realized  FIG 

when   we    remember   that   only 

those  planes  that  are  parallel  to  each  other  can  be  equally  inclined 
to  a  given  plane,  hence  planes  that  make  different  angles  with  a 
given  plane  cannot  be  parallel  to  one  another.  Moreover,  since  in 
crystallography  planes  are  distinguished  from  each  other  by  their 
inclinations  to  standard  planes,  it  follows  that  planes  that  make 
different  angles  with  a  given  plane  are  different.  It  further  fol- 
lows that  when  two  planes  on  one  crystal  have  a  different  inter- 
facial angle  than  that  possessed  by  two  planes  on  a  second  crystal, 
the  planes  on  the  two  crystals  cannot  be  the  same. 

The  values  of  interfacial  angles  may  thus  serve  to  distinguish 
between  substances  that  are  otherwise  apparently  identical.  For 
instance,  certain  light  Jm>wn  crystals  of  calcite  (CaCO3)  bear  a 
strong  resemblance  to  certain  crystals  of  light  brown  siderite 
(FeCO3).  If  the  crystals  are  six-sided  (rhombohedrons)  they 
contain  corresponding  interfacial  angles,  which  on  the  calcite 
have  a  value  of  106°  15',  and  on  the  siderite  107°.  By  careful 
measurement  this  difference  may  be  recognized  and  the  two  min- 
erals may  be  distinguished. 


CHAPTER  II. 


THE  LAW  OF  THE  CONSTANCY  OF  INTERFACIAL  ANGLES. 

Statement  of  the  Law. — The  observation  and  comparison 
of  crystals  have  shown  that  however  much  the  crystals  of  a  given 
substance  may  vary  in  size  and  shape,  their  corresponding  interfacial 
angles  are  the  same  in  value,  provided  they  be  measured  under 
the  same  conditions. 

Significance  of  the  Law.— This  law  expresses  the  fact  that  it 
is  not  the  general  shape,  or  habit,  of  a  crystal  that  is  its  most 
important  external  feature,  but  the  inclination  of  the  planes  by 
which  it  is  bounded.  Two  crystals  may  look  very  much  alike, 
i.e.,  they  may  be  similar  in  habit,  but  if  the  inclinations  of  their 


FIG.  ii. — Three  crystals  bounded  by  the  same  planes,  but  with  different  habits. 

corresponding  faces  are  different  the  crystals  themselves  are 
made  up  of  different  crystallographic  elements.  On  the  other 
hand,  if  two  crystals  that  are  apparently  entirely  unlike  each 
other  are  bounded  by  planes  with  their  corresponding  interfacial 
angles  equal  in  value,  the  corresponding  crystal  faces  are  regarded 
as  the  same  faces  and  the  crystals  are  crystallographically 
identical. 

The  crystals  A,B,  and  C  represented  in  the  subjoined  figure 
(Fig.  u)  are  very  different  in  habit,  and  yet  they  are  bounded 

14 


THE    LAW    OF    THE    CONSTANCY    OF   INTERFACIAL  ANGLES       15 

by  the  same  planes,  since  the  interfacial  angles  on  A  are  identical 
in  value  with  the  corresponding  angles  on  B  and  C  (between 
planes  marked  with  the  same  letters). 

The  condition  imposed  by  the  law,  viz.,  that  the  crystals  shall 
"be  measured  under  the  same  conditions,"  is  necessitated  by  the 
fact  that  crystals  deport  themselves  toward  physical  agencies  in  a 
different  manner  from  non-crystallized  substances.  Under  the 
influence  of  changes  in  temperature,  for  instance,  they  may 
expand  or  contract  differently  in  different  directions,  and  the 
inclinations  of  their  faces  will  vary 
accordingly.  Even  the  same  crystal 
will  yield  different  values  for  the 
same  interfacial  angle  when  this  is 
measured  at  different  temperatures, 
hence  in  comparing  the  angles  on 
different  crystals  it  is  necessary  to 
know  at  what  temperature  they  were 
measured  in  order  to  decide  whether 
or  not  the  faces  including  the  angles 
are  the  same. 

The  expression  of  the  law  further  indicates  that  it  applies 
only  to  crystals  of  the  same  substance.  From  this  we  may 
rightly  infer  that  crystals  of  different  substances  possess  different 
interfacial  angles  even  though  bounded  by  similar  crystallographic 
planes.  As  a  fact,  except  among  the  bodies  crystallizing  in 
what  is  known  as  the  regular  or  isometric  system,  and  with  the 
exception  of  a  very  few  angles  on  other  crystallized  bodies,  the 
crystals  of  different  substances  possess  different  interfacial 
angles,  which,  however,  are  so  distinctive  for  each  substance  that 
they  may  often  serve  as  a  means  for  its  identification.  The 
values  of  the  interfacial  angles  are  therefore  characteristic  for 
different  substances.  They  must  consequently  vary  according  as 
the  chemical  composition  of  the  substance  varies.  When  the 
direction  and  amount  of  the  variation  caused  in  the  value  of  any 
angle  by  the  admixture  of  a  certain  chemical  element  into  the 
molecule  is  known  the  quantity  of  this  element  introduced  may 
frequently  be  determined  by  the  difference  in  the  value  of  the 


l6  GEOMETRICAL    CRYSTALLOGRAPHY 

angle  observed  and  that  of  the  corresponding  angle  in  the  pure 
substance  (see  pages  231-232). 

In  the  mineral  calcite  (CaCO3)  the  angle  between  the  faces 
R  and  R'  is  74°  55'  (Fig.  12),  while  in  magnesite  (MgCO3),  which 
crystallizes  like  calcite,  it  is  72°  31'.  In  the  mineral  dolomite 
(MgCaCO3),  which  crystallizes  in  the  same  shape,  the  corre- 
sponding angle  is  73°  45'.  This  is  between  that  of  calcite  and 
that  of  magnesite. 

Morphotropism. — The  influence  which  the  introduction  of 
an  element  or  a  group  of  elements  into  a  compound,  or  their 
abstraction  from  a  compound,  exerts  upon  the  values  of  the 
interfacial  angles  is  known  as  morphotropism.  The  action  itself 
is  known  as  morphotropic  action.  The  morphotropic  action  of 
Mg  in  calcite,  for  instance,  is  to  decrease  the  value  of  the  angle 
RAR'. 

Corollary  to  the  Law. — Since  the  interfacial  angles  are  the 
important  elements  in  defining  the  nature  of  crystals,  and  not 
the  sizes  or  shapes  of  the  faces  or  their  distances  from  the  centers 
of  the  crystals,  and  since  the  only  planes  that  can  make  equal 
interfacial  angles  with  a  given  plane  are  those  that  are  parallel 
to  each  other,  it  follows  that  parallel  faces  in  crystallography 
may  be  regarded  as  the  same  face,  situated  at  different  distances 
from  the  center  of  the  crystal. 

Thus  a  small  cube  and  a  large  cube  are  crystallographically 
identical  forms,  which  differ  from  each  other  only  in  the  distances 
of  their  bounding  planes  from  the  centers  of  the  respective  forms. 


CHAPTER  III. 


THE  LAW  OF  SIMPLE  MATHEMATICAL  RATIOS. 

The  Designation  of  the  Positions  of  Planes  in  Space.— 

The  position  of  a  plane  in  space  may  be  denned  by  referring  it  to  a 
system  of  three  lines  intersecting  at  a  common  point,  just  as  the 
position  of  a  line  on  a  plane  may  be  defined  by  referring  it  to  two 
other  intersecting  lines. 

Planes  on  crystals  are  referred  to  a  system  of  imaginary  lines, 
known  as  axes,  which  are  supposed  to  intersect  at  the  center  of 
the  crystal.  The  positions  and  inclinations  of  the  planes  are 


-b- 


-fb 


FIG.   13. — System  of  axes. 


FIG.  14. 


defined  by  expressing  the  relations  between  the  distances  at  which 
they  cut  the  axes,  measured  from  the  point  of  their  common 
intersection. 

In  figure  13  is  illustrated  a  scheme  of  axes  to  which  the  planes 
on  certain  crystals  may  be  referred.  In  order  to  locate  the  planes 
accurately  each  axis  is  designated  by  a  letter,  and  the  two  ends 
of  each  axis  are  distinguished  by  different  signs.  The  positions 
of  planes  are  thus  easily  indicated  by  stating  the  relative  distances 
at  which  they  cut  the  three  axes.  The  plane  ABC  (Fig.  14)  may 
be  defined  as  a  plane  cutting  the  a  axis  at  the  distance  OA,  the  b 

17 


1 8  GEOMETRICAL   CRYSTALLOGRAPHY 

I 

axis  at  OB,  and  the  c  axis  at  OC.  The  plane  HKL  may  likewise 
be  defined  as  a  plane  cutting  a  at  OH,  b  at  OK,  and  c  at  OL. 

Parameters. — In  all  measurements  of  distance  a  standard 
length  is  assumed,  in  terms  of  which  the  result  of  the  measurement 
is  expressed.  In  ordinary  practice  the  standards  made  use  of 
are  the  inch,  foot,  yard,  mile,  or  metre,  kilometre,  etc.  In 
crystallography  the  standard  lengths  are  those  distances  at  which 

"  a  selected  plane,  called  the  groundplane  or  groundform,  cuts, 
or  intercepts,  the  axes.  These  distances  are  taken  as  units,  and 
the  distances  at  which  other  planes  on  the  same  crystal  intercept 
the  corresponding  axes  are  expressed  in  terms  of  these  units. 

-  The  resulting  expressions  are  known  as  the  parameters  of  the 
plane.  If  ABC  in  figure  14  is  taken  as  a  groundplane,  then  its 
intercepts  OA,  OB,  and  OC  on  the  three  axes  are  the  unit  lengths 
on  these  axes.  The  plane  HKL  intercepts  the  axes  at  OH,  OK, 
and  OL.  If  OH- 2  OA;  OK  =  OB,  and  OL  ==  1/2  OC,  then 
the  position  of  HKL  may  be  expressed  by  stating  that  its  intercept 
on  a  is  twice  the  unit  length  on  a,  on  b  the  intercept  is  the  unit  on 
b,  and  on  c  the  intercept  is  1/2  the  unit  on  c,  or,  briefly,  the  symbol 
representing  the  plane  is  2a  :  b  :  i/2C,  in  which  a,  b,  and  c  repre- 
sent the  unit  lengths  on  the  three  axes.  The  values  2,  i,  and  1/2 
are  the  parameters  of  the  plane  on  the  axes  a,  b,  c,  respectively. 

General  Symbol  for  a  Plane. — The  most  general  symbol  for 
any  plane  is  ma  :  nb  :  pc,  in  which  m,  n,  and  p  represent  definite 
values  other  than  one.  A  plane  with  this  symbol  cuts  the  three 
axes  at  different  distances,  and  no  one  of  the  distances  is  the  unity 
for  that  axis.  But  parallel  planes  in  crystallography  are  regarded 
as  the  same  plane  (see  page  16),  hence  one  may  imagine  them 
moved  parallel  to  themselves  without  affecting  their  relations  to 
other  planes.  The  plane  ma,  nb,  pc,  may  be  shifted  in  position 

m  p 

until  it  cuts  b  at  unity.  It  will  then  cut  a  at  —a,  and  c  at  -c,  pro- 
vided its  new  position  is  parallel  to  its  original  one,  and  the 
symbol  of  the  plane  will  become  m'a  :  b  :  p'c.  The  parameters  are 
thus  ratios  between  the  intercepts  of  a  plane  on  the  axes  and  the 
unity  distances  on  the  same  axes,  when  the  plane  is  moved  paral- 
lel to  itself  until  it  intercepts  one  of  the  axes  at  a  unity  (or 


THE   LAW   OF   SIMPLE   MATHEMATICAL   RATIOS 


standard)  distance.  By  this  method  corresponding  planes  of 
similar  crystals  are  represented  by  the  same  symbol  whether  the 
crystal  be  small  or  large.  The  parameters  are  ratios  and  not 
absolute  distances. 

One  of  the  theorems  of  solid  geometry  states  that  if  two  paral- 
lel planes  are  cut  by  any  third  plane,  the  intersections  are  parallel. 
Let  ABC  (Fig.  15)  be  a  plane 
cutting  the  axes  a,  b,  and  c  at 
A,  B,  and  C,  respectively, 
and  let  OA  =  w,  O~B=n,  and 
OC  =  #.  Let  A',  B',  C'  be  the 
position  of  this  plane  when  it 
is  moved  parallel  to  itself  until 
it  cuts  b  at  unity  (B7).  Pass 
a  plane  through  the  axes  OC' 
(c)  and  OB7  (6),  then  will  the 
intersections  of  this  plane  with 


FIG.  15. 


ABC  and  A'B'C',  or  the  lines 

BC  and  B'C',  be  parallel.   The 

triangles  OBC  and  OB'C'  are  therefore  similar,  and  OB  :  OB' 

:  :  OC  :  OC'=n  :  i  :  :  p  :  OC',  and  OC'  (the  intercept  of  A'B'C' 

on  c)  =5-.     In  a  similar  way  it  can  be  proven  that  the  new  inter- 
n 

m 

cept  on  a  =  —  . 
n 

The  values  m,  n,  and  p  of  the  first  symbol  are  not  the  same  as 
m',  i,  and  p'  of  the  second  symbol,  but  the  ratios  between  m,  n,  and 
p  are  the  same  as  those  between  mf,  i  ,  and  p'.  This  being  the  case, 
it  is  convenient  to  express  the  symbol  of  a  plane  in  the  form  of  a 
ratio,  for  then  it  becomes  a  general  symbol  for  any  plane  parallel 
to  a  given  plane.  Thus  m'a  :  b  :  p'c  is  the  symbol  for  any  plane 
cutting  the  three  axes  in  the  ratio  m'  times,  once,  and  p'  times  the 
unity  distances  on  a,  b,  and  c. 

For  instance,  the  symbol  2a  :  b  :  i/2C  designates  any  plane 
that  cuts  the  axes  at  distances  that  are  in  the  ratio  of  2  :  i  :  1/2 
times  the  unities  on  the  three  axes.  The  symbols  40  :  2b  :  c  and 
a  :  i/2b  :  1/40  designate  the  same  plane  as  does  the  symbol 


20  GEOMETRICAL   CRYSTALLOGRAPHY 

2a  :  b  :  i/2C  because  the  ratios  between  the  parameters  are  the 
same  in  all  cases.  The  plane  40  :  2b  :  c  may  be  regarded  as  be- 
ing at  twice  the  distance  from  the  center  of  the  crystal  as  is  the 
plane  2a  :  b  :  i/2C,  and  the  plane  a  :  1/26  :  1/40  at  only  half  this 
distance;  but  all  the  planes  make  the  same  interfacial  angles  with 
the  corresponding  adjacent  planes  and  hence  are  regarded  as  the 
same  plane.  If  the  three  symbols  express  any  difference  in  the 
planes  it  is  simply  with  respect  to  their  actual  distance  from  the 
center  of  the  crystals  on  which  they  are  found;  or,  in  other  words, 
they  express  simply  a  difference  in  the  size  of  the  crystals.  A 
cube  is  a  cube,  whether  it  measures  i/io  of  an  inch  or  10  inches  on 
an  edge.  Crystallography  takes  no  account  of  the  size  of  crystals; 
it  deals  only  with  the  distribution  and  inclination  of  the  planes 
that  compose  them. 

Crystallographic  Notation. — In  discussing  crystals  and 
crystal  planes  it  is  necessary  to  make  use  of  some  method  of 
representing  them  accurately  and  at  the  same  time  graphically. 
The  principles  upon  which  one  system  of  notation  is  based  are 
explained  in  the  previous  paragraph.  This  is  known  as  the 
system  of  parameters  or  the  Weiss  system,  after  its  inventor. 

The  Weiss  System  of  Notation. — In  this  system  the  crys- 
tallographic  axes  are  invariably  written  in  the  order  a  :  b  :  c, 
and  to  these  are  prefixed  the  parameters,  when  these  vary  from 
unity.  The  symbol  a  :  b  :  c  without  any  parameters  always 
refers  to  a  plane  of  the  groundform,  that  is,  to  a  plane  cutting  the 
three  axes  at  the  distances  assumed  as  the  unities.  The  paral- 
lelism of  a  plane  to  a  crystallographic  axis  is  indicated  by  the  sign 
of  infinity,  co ,  written  in  its  proper  place  as  a  parameter.  Thus 
oca  :  b  :  c,  represents  a  plane^arallel  to  the  a  axis  and  cutting  b 
and  c  at  unity,  oca  :  co&  :  c,  is  a  plane  cutting  c  at  unity,  and  at 
the  same  time  parallel  to  a  and  b.  Further,  it  is  customary  for 
simplicity's  sake  to  make  the  parameter  on  either  the  a  or  the  b 
axis  unity,  and  to  reduce  the  symbol  accordingly.  For  instance, 
if  the  measurement  of  a  crystal  should  yield  420  :  6b  :  I2C  as  the 
symbol  for  a  certain  plane,  this  would  be  reduced  to  the  form 
70  :  b  :  2C,  which  would  become  the  symbol  for  the  plane. 

By  reference  to  figure   13    (page  17)  it  will  be  noted  that  this 


THE    LAW    OF    SIMPLE    MATHEMATICAL   RATIOS  21 

symbol  stands  for  a  plane  in  the  upper  right-hand  division  or  oc- 
tant of  space  included  within  the  halves  of  the  three  axes.  Corre- 
sponding planes  in  the  other  octants  on  the  upper  half  of  the 
crystal  would  be  represented  by  —  'ja  :  b  :  20;  —  ja:  —b:2C,  and 
7<z  :  —  b  :  2C.  On  the  lower  half  of  the  crystal  we  should  have 
the  corresponding  planes  ja  :  b  :  —  20;  —  ja:  —b  :  —  2c; 
—  7  a  :  b  :  —2C,  and  70  :  —  b  :  —  20.  Each  symbol  thus  represents 
a  different  plane,  the  position  of  which  on  the  crystal  is  indicated 
by  the  signs.  Except  with  respect  to  the  signs,  all  the  symbols 
are  alike.  Hence,  if  the  presence  of  one  of  the  planes  should 
necessitate  the  presence  of  all  the  others,  then  the  signs  could 
be  omitted  without  affecting  the  significance  of  the  symbol.  It 
would  then  indicate  not  simply  a  single  one  of  these  correspond- 
ing planes,  but  any  one  of  them,  that  is,  it  would  become  a 
general  symbol  for  all.  Upon  further  study  it  will  be  found 
that  the  presence  of  a  given  plane  does  necessitate  the  presence 
of  one  or  more  similar  planes  on  the  same  crystal,  and  in  this 
case  the  symbol  of  any  one  of  the  planes,  written  without  signs, 
becomes  the  symbol  for  all  of  them. 

Crystal  Forms. — A  crystal  form  is  the  sum  of  all  the  planes  on 
a  crystal  that  may  be  represented  by  symbols  differing  only  in 
signs.  The  eight  planes  indicated  in  the  last  paragraph,  and  of 
which  70  :  b  :  20  is  one,  constitute  a  form. 

Naumann's  System  of  Notation. — The  Weiss  system  of 
notation  has  to  do  primarily  with  the  planes  comprising  crystal 
forms.  It  is  cumbersome  and  more  or  less  indefinite,  as  a  sym- 
bol may  be  not  merely  the  symbol  of  a  form,  but  the  symbol  of 
one  of  the  faces  of  the  form  as  well.  Naumann  has  proposed  a 
system  of  notation  which  is  in  principle  the  same  as  that  of  Weiss, 
but  which  has  the  advantage  of  representing  forms  while  at  the 
same  time  it  is  briefer  than  that  of  Weiss. 

In  this  scheme  the  groundform  is  represented  by  the  letters  O 
or  P,  according  as  the  unities  on  the  three  axes  are  equal  or 
different  in  value,  and  the  parameters  of  the  planes  are  written 
before  or  after  the  symbol  for  the  groundform,  according  as  they 
refer  to  the  c  or  to  one  of  the  other  axes.  One  of  the  latter  is 


22  GEOMETRICAL   CRYSTALLOGRAPHY 

always    made    unity,    and   unity   values   are  omitted   from   the 
symbol. 

The  symbol  a  :  b  :  c  becomes  in  the  Naumann  notation  O  or  P, 
20,  :  b  :  i/2C  becomes  1/2^2  or  1/202,  and  ja  :  b  :  20  becomes 
2?7  or  207.  The  symbol  a  :  $b  :  20  would  become  2?5  or 
205,  a  :  b  :  2C  would  be  2?  or  2O,  a  :  2b  :  c  or  20,  :  b  :  c  would 
be  P2  or  O2.  The  ocPoo  in  Naumann's  notation  is  oo#  :  b  :  &c 
or  a  :  oo£  :  oo c  of  Weiss,  ooa  :  oo&  :  c  is  represented  by  oP, 
for  to  make  a  or  b  unity  one  must  divide  by  GO,  when  we  have 

oo          oo          i 

— a  :  — b  :  — c,  which  is  la  :  ib  :  oc. 

00  00  00 

Dana's  Notation. — Prof.  Dana  has  further  simplified 
Naumann's  system  by  substituting  a  hyphen  for  the  initial  O  or 
P,  and  the  letter  i  or  I  for  the  infinity  sign  oo.  1/2? 2  of  Naumann 
=  1/2—2  of  Dana.  2Py  =  2—  7,  oop2  =  i  —  2,  2Poo  =  2— i, 
ooP  oo  =i  — i.  When  the  groundform  exists  alone  it  is  represented 
by  i.  Thus  O  or  P  of  Naumann  =  i  of  Dana. 

If  only  one  parameter  differs  from  unity  it  is  written  alone 
if  it  refers  to  c,  as  2P  =  2,  P=i;  but  if  it  refers  to  a  or  b,  it  is 
preceded  by  i,  as  P2  =  i  —2,  P  oo  =  i  — i. 

Statement  of  the  Law  of  Simple  Mathematical  Ratios.— 
The  second  law  of  crystallography  may  be  expressed  as  follows: 
Experience  shows  that  only  those  planes  occur  on  crystals  whose 
parameters  are  either  infinite  or  are  small  fractions  or  small  even 
multiples  of  unity.  The  ratio  existing  between  them  is  simple. 

Under  certain  circumstances  of  growth  not  yet  clearly  under- 
stood there  occur  on  some  crystals  small  planes  with  very  large 
and  sometimes  irrational  parameters.  These  are  known  as 
vicinal  planes.  They  are  probably  produced  by  causes  which 
are  partly  or  wholly  external  in  origin,  and  are,  strictly  speaking, 
not  idiomorphic  forms  due  to  the  crystallizing  force  inherent  in 
the  molecules  of  the  crystallizing  substance. 

If  this  law  is  a  correct  statement  of  fact,  the  plane  i .  675  a  :  b 
:  3.272  c  is  crystallographically  impossible.  In  the  experience 
of  crystallographers  no  plane  with  these  parameters  has  been 
found,  unless  it  may  be  in  some  of  the  vicinal  forms. 

Practical  Utility  of  the  Law.— The  practical  value  of  the 


THE    LAW    OF    SIMPLE    MATHEMATICAL   RATIOS  23 

law  is  soon  seen  when  we  come  to  make  use  of  it  in  determining 
the  symbols  of  planes  from  the  measurements  of  their  interfacial 
angles.  The  calculation  of  the  intercepts  of  a  plane  is  apt  to 
lead  to  results  that  vary  slightly  from  simple  fractions  or  small 
whole  numbers,  on  account  of  the  practical  impossibility  of 
determining  accurately  the  true  values  of  the  interfacial  angles 
because  of  the  imperfections  of  our  instruments  and  the  irregu- 
larities in  the  surfaces  of  the  crystal  faces.  Under  these  cir- 
cumstances we  confidently  take  the  nearest  simple  fractions  or 
the  nearest  whole  numbers  to  represent  the  true  intercepts  of  the 
plane  under  investigation.  For  example,  if  we  should  calculate 
the  intercepts  of  a  certain  plane  to  be  i .  973  a  :  b  :  .  508  c,  we 
would  not  hesitate  to  write  its  symbol  20,  :  b  :  1/2  c. 

The  Miller  System  of  Notation.— There  is  another  system 
for  designating  crystal  planes  that  is  even  more  widely  used  than 
the  system  of  Naumann  or  of  Dana.  Its  advantage  over  these 
latter  is  due  largely  to  the  greater  ease  with  which  the  symbols 
can  be  manipulated  in  mathematical  discussions.  Although 
very  simple  in  principle,  it  is  a  little  more  difficult  of  comprehension 
than  Naumann's  system. 

In  the  Miller  system  the  symbol  of  a  plane  consists  of  its  indices 
written  in  the  order  given  above  for  the  axes,  viz.,  a  :  b  :  c,  and  in 
the  simplest  form  possible,  without  the  use  of  fractions.  The 
indices  are  the  reciprocals  of  the  parameters.  The  symbol  3?3/2 
of  Naumann  (when  3/2  refers  to  the  a  axis)  represents  a  plane 
whose  parameters  for  the  different  axes  are  3/20  :  b  :  $c.  The 
reciprocals  of  these  parameters  are  2/3  :  i  :  1/3  taken  in  order. 
When  cleared  of  fractions  these  become  2:3:1,  and  the  symbol 
of  the  plane,  according  to  the  Miller  system,  is  2  3  i.  The  indices 
are  always  written  in  the  same  order  and  the  symbol  for  the  axes 
is  omitted. 

The  transfer  between  the  Naumann  and  the  Miller  symbols 
is  easily  effected  if  it  is  remembered  that  the  latter  are  always 
written  in  the  order  a  :  b  :  c  and  that  no  fractions  are  employed. 
In  the  Naumann  symbol,  on  the  other  hand,  the  parameter  on  c 
is  written  before  the  initial  representing  the  groundform  of  the 
crystallized  substance,  except  in  the  isometric  system  in  which 


24  GEOMETRICAL    CRYSTALLOGRAPHY 

there  is  no  distinction  made  between  the  three  axes.  In  trans- 
ferring from  the  Naumann  to  the  Miller  system,  then,  it  is  only 
necessary  to  write  the  parameters  in  the  a  :  b  :  c  order,  take  their 
reciprocals  and  clear  of  fractions.  To  transfer  in  the  opposite 
direction  the  reverse  process  is  employed,  the  smaller  of  the 
parameters  on  a  and  b  being  made  unity.  When  the  parameters 
are  obtained,  only  those  not  unity  are  represented,  and  that 
referring  to  the  c  axis  is  written  in  front  of  the  initial  that  stands 
for  the  groundform. 


Thus: 

3/2  P  3*=  30  :  b  :  3/2  c 

i/3    i     2/3 

(parameters) 
(indices) 

i     3       2 

(Miller  symbol) 

3  P  3/2*=  3/2<*  :  b  :  $c 

(parameters) 

2/3        i     i/3 

(indices) 

2          3       i 

(Miller  symbol) 

3  P  3/2*=.  a  :  3/26  »^c 

(parameters) 

1     2/3       1/3 

(indices) 

3       2          i 

(Miller  symbol) 

and  364=       30  :  66  :  40 

(indices) 

1/30,  :  1/66  :  i/4c 

(parameters) 

2a  :  b  :  3/26 

(parameter  on  b  ==  i) 

3/2  P  2* 

(Naumann  symbol) 

010=         oa  :  b  :  oc 

(indices) 

v>a  :  b  :  ooc 

(parameters) 

ooP  c** 

(Naumann  symbol) 

*  In  the  Naumann  symbols  the  signs  ~  and  ••  refer  to  the  a  axis  and  the  sign  - 
to  the  b  axis.  When  no  sign  is  used  above  the  parameter  after  the  initial  letter  the 
parameter  refers  to  the  a  and  the  6  axes  indifferently. 


CHAPTER  IV. 

SYMMETRY. 

Symmetry. — By  the  symmetry  of  a  body  is  meant  the  regular 
arrangement  of  its  parts  with  respect  to  planes,  lines,  or  points.  Four 
types  of  symmetry  are  recognizable  in  crystals,  of  which  three  are 
much  more  important  than  the  fourth.  These  three  are: 

1.  Symmetry  with  reference  to  planes. 

2.  Symmetry  with  respect  to  lines. 

3.  Symmetry  with  respect  to  points. 

Symmetry  with  Respect  to  Planes.-^ When  two  halves  of  a 
body  bear  to  each  other  the  relation  of  an  object  and  its  image, 
the  two  halves  are  said  to  be  symmetrical  about 
the  plane  that  divides  them.  When  the  body 
possesses  only  one  such  plane  of  symmetry,  it 
is  described  as  being  symmetrical  about  a  single 
plane.  (See  Fig.  16.)  When  it  possesses  three 
such  planes,  it  is  symmetrical  about  three 
planes.  The  planes  are  known  as  planes  of 
symmetry. 

Symmetry  with  Respect  to  Lines.  — 
Many  bodies  may  be  revolved  about  a  line  so 
situated  within  it  that  a  revolution  through  an 

FIG.       16.—  Body 
angle     less    than    360      Will    Cause    its    parts    to    divided  by  one  plane 

occupy  successively  the  same  relative  positions  of  symmetry- 
as  they  originally  occupied.  When  such  a  condition  is  brought 
about  by  a  revolution  through  180°,  or  twice  during  a  complete 
revolution,  the  body  is  said  to  possess  a  twofold  or  binary  sym- 
metry. When  brought  about  by  revolutions  of  120°,  90°,  or  60°, 
it  is  spoken  of  as  possessing  threefold  or  ternary,  fourfold  or 
quadratic,  and  sixfold  or  hexagonal  symmetry.  The  line  about 
which  such  a  revolution  is  possible  is  called  an  axis  of  symmetry. 
The  axes  of  twofold  symmetry  are  designated  as  secondary 

25 


26  GEOMETRICAL   CRYSTALLOGRAPHY 

axes  of  symmetry;  those  of  more  than  twofold  symmetry  are 
designated  as  principal  axes  of  symmetry. 

Symmetry  axes  are  polar  when  their  opposite  ends  terminate 
differently;  i.e.,  in  crystals,  when  their  opposite  ends  terminate  in 
different  planes,  edges,  or  solid  angles  (compare  Fig.  33). 

Symmetry  with  Reference  to  Points. — Bodies  are  sym- 
metrical about  a  point  when  a  straight  line  drawn  through  this 
point  terminates  at  equal  distances  on  its  opposite  sides  in 
similar  surfaces  or  angles.  Such  a  point  is  called  a  center  of 
symmetry.  A  crystal  possessing  a  center  of  symmetry  must  be 
bounded  by  pairs  of  parallel  planes  (see  Figs.  200  and  201). 

A  sphere  and  a  cube  are  symmetrical  about  a  point  at  their 
centers.  A  right  cone  with  a  circular  base  and  a  right  pyramid 
with  a  square  base  are  symmetrical  about  lines  passing  through 
their  apices  and  the  centers  of  their  bases.  The  latter  possesses 
a  fourfold  symmetry  and  has  four  vertical  planes  of  symmetry, 
two  passing  through  the  corners  of  the  base  and  two  through  the 
centers  of  its  sides.  A  right  pyramid  with  a  rectangular  base  that 
is  not  a  square  possesses  two  vertical  planes  of  symmetry  through 
the  centers  of  the  sides  of  its  base.  A  right  pyramid  with  an 
isosceles  triangle  as  its  base  possesses  a  single  plane  of  symmetry 
passing  vertically  through  its  base  and  bisecting  the  angle  between 
its  equal  sides. 

Symmetry  in  Crystallography. — It  has  already  been  re- 
marked that  the  inclinations  of  the  planes  on  a  crystal  are  of  much 
more  importance  than  their  distances  from  its  center.  Similarly 
the  notion  of  symmetry  in  crystals  applies  to  the  directions  of  its 
planes  rather  than  to  the  distances  of  these  planes  from  the  plane 
of  symmetry. 

Crystallographically  the  two  figures  on  the  next  page  (Fig.  17) 
are  symmetrical  about  the  plane  AB  passed  through  them  per- 
pendicular to  the  plane  of  the  paper.  The  two  parts  of  figure  17  A 
are  symmetrical  not  only  with  respect  to  the  inclinations  of  their 
bounding  lines,  but  also  with  respect  to  the  distances  of  these  lines 
from  the  plane  of  symmetry.  This  polygon  is  ideal  in  its  sym- 
metry. Figure  17  B  is  symmetrical  with  respect  to  the  in- 
clination of  the  lines  on  opposite  sides  of  the  plane  AB,  but 


SYMMETRY 


not  with  respect  to  the  distances  of  these  lines  from  the  plane. 
Every  line  on  either  side  of  the  plane  of  symmetry  corresponds  to 
a  line  on  the  opposite  side,  and  the  corresponding  lines  on  each 
side  make  equal  angles  with  the  corresponding  adjacent  lines. 
The  angles  a,  b,  c  are  equal,  respectively,  to  a',  b',  c'. 


u 

A  B 

FIG.   17. — Diagrams  illustrating  symmetry. 

In  the  same  way  the  crystals  B  and  C  in  figure  18  are  sym- 
metrical in  the  same  degree  as  is  the  crystal  A.  But  the  latter 
crystal  is  ideal  in  its  symmetry,  while  the  other  two  are  distorted 
through  irregularities  in  their  growth. 

Crystal  Forms. — One  definition  of  a  crystal  form  has  been 


FIG.  18.  —  Three  crystals  bounded  by  the  same  planes,  but  with  different  habits. 


given  (page  21).  It  may  better  be  defined  as  the  sum  of  all 
the  planes  demanded  by  a  crystal's  symmetry,  in  consequence 
of  the  presence  of  one.  It  is  for  this  reason  that  the  symbol  of 
a  single  plane  of  a  form  may  be  made  to  stand  for  the  entire  form. 
Occasionally  through  accidents  of  growth  one  or  more  planes 
of  a  form  may  be  crowded  from  the  crystal.  Plane  z  in  the  upper 


28  GEOMETRICAL    CRYSTALLOGRAPHY 

left-hand  corner  of  crystal  C,  figure  18,  for  instance,  is  very  small. 
Continued  growth  of  P  and  r  might  have  caused  it  to  disappear 
completely.  An  accident  of  this  kind  is  not  recognized  in  the 
symbol  of  a  crystal,  though  the  absence  of  the  plane  is  usually 
mentioned  in  the  description  of  the  crystal's  habit. 

Grades  of  Symmetry. — Differences  in  the  number  of  planes, 
axes,  and  centers  of  symmetry  possessed  by  crystal  forms  deter- 
mine their  grade  of  symmetry — the  greater  the  number  of  elements 
of  symmetry  present,  the  higher  the  grade  of  symmetry. 

Law  of  Symmetry. — Each  crystallizing  substance  possesses 
a  characteristic  grade  of  symmetry.  Only  those  forms  may  occur 
on  its  crystals  that  possess  the  same  grade  of  symmetry. 

Crystallographic  Systems. — All  crystal  forms  possessing  the 
same  grade  of  symmetry  and  all  forms  that  may  be  regarded  as 
derived  from  these  by  the  suppression  of  a  certain  definite  propor- 
tion of  their  planes  in  accordance  with  certain  definite  laws  (par- 
tial forms)  are  grouped  together  as  a  system. 

Two .  Kinds  of  Planes  of  Symmetry. — Thus  far  we  have 
discussed  planes  of  symmetry  without  distinguishing  between  the 
different  kinds.     As  a  matter  of  fact,  two 
kinds  must  be  distinguished — principal  and 
secondary  planes. 

Principal  planes  of  symmetry  contain  two 
or  more  equivalent  and  interchangeable 
directions — directions  that  may  be  imagined 
as  interchanged,  without  affecting  the  shape 
of  the  crystal  through  which  they  pass.  They 
are  symmetry  planes  to  which  two  or  more 
other  symmetry  planes  are  perpendicular. 
FIG.  1 9.  ^-Crystal  Secondary  planes  of  symmetry  possess  no 

possessing  principal  and    . 

secondary     planes     of  interchangeable  directions. 

In  the  form  represented  by  figure  19,  the 

plane  a, a',  —a,  —a7,  is  a  principal  plane  of  symmetry,  while  the 
plane  a,c,  —a,  — c  is  a  secondary  plane.  The  directions  a  —a, 
and  a'  —a'  may  be  interchanged  by  a  horizontal  revolution  of  the 
crystal  through  an  arc  of  90°  without  affecting  the  shape  of  the 
crystal  in  the  least.  On  the  other  hand,  the  directions  repre- 


SYMMETRY  2Q 

sented  by  the  lines  a  — aandc  — c  are  not  equivalent  directions, 
because  if  the  crystal  were  revolved  about  a'  —a'  through  an  arc 
of  90°  until  c  —  c'  takes  the  direction  now  held  by  a  —a, 
its  long  axis  would  be  horizontal,  whereas  originally  it  was 
vertical.* 

Number  and  Characterization  of  Crystal  Systems. — It  . 
can  be  proven  mathematically  that  all  the  crystal  forms  that  are 
possible  may  be  divided  into  32  groups  or  classes,  each  charac-  ' 
terized  by  a  special  grade  of  symmetry.  To  each  class  a  dis- 
tinctive name  is  given  which  indicates  its  grade  of  symmetry.  It 
can  also  be  shown  that  of  these  groups  there  are  six  from  which 
the  other  26  may  be  considered  as  derived.  There  are,  therefore, 
six  systems  into  which  all  crystals  may  be  grouped.  Each 
system  includes,  therefore,  not  only  the  forms  possessing  a  certain 
grade  of  symmetry,  but  also  all  forms  that  may  be  considered  as 
derived  from  these;  i.e.,  all  forms  that  may  be  referred  to  Nthe 
same  system  of  coordinates  or  axes.  The  groups  or  systems 
generally  recognized  are  as  follows : 

System  Forms 

Isometric With  3  principal  and  6  secondary  planes  of 

symmetry,    and   partial    forms    derived    from 

these. 
Hexagonal With  i  principal  and  6  secondary  planes  of 

symmetry,    and   partial   forms   derived   from 

these. 
Tetragonal  .- With  i  principal  and  4  secondary  planes  of 

symmetry,    and   partial   forms    derived   from 

these. 
Orthorhombic ....  With  o  principal  and  3  secondary  planes  of 

symmetry,    and   partial    forms    derived    from 

these. 

*  It  is  absolutely  necessary  that  the  student  should  understand  the  distinction 
between  principal  and  secondary  planes  of  symmetry  before  proceeding  further. 
An  excellent  exercise  for  this  purpose  is  to  separate  a  number  of  crystal  models, 
such  as  are  recommended  in  the  footnote,  page  n,  into  groups  in  accordance  with 
the  crystal  system  represented  by  them. 


30  GEOMETRICAL   CRYSTALLOGRAPHY 

Monoclinic With  o  principal  and   i   secondary  plane  of 

symmetry,   and   partial   forms   derived   from 

these. 
Triclinic With  no  plane  of  symmetry,  but  only  a  center 

of  symmetry,  and  partial  forms  derived  from 

these. 


CHAPTER  V. 
THE  ISOMETRIC  OR  REGULAR  SYSTEM. 

Division  of  the  System. — The  forms  belonging  to  the  iso- 
metric system  like  those  belonging  to  all  the  other  systems  may 
be  separated  into  two  groups,  viz.,  holohedral  and  partial  forms. 

The  holohedral  (oAos,  whole,  and  efyxx,  face)  forms  are 
those  possessing  all  the  planes  demanded  by  the  complete  sym- 
metry of  the  system.  A  form  of  this  kind  is  known  as  a  holohedron. 
The  partial  forms  are  made  up  of  the  same  planes  as  the  corre- 
sponding holohedral  forms,  but  some  of  those  demanded  by  the 
complete  symmetry  of  the  system  are  not  present.  Partial  forms 
are  of  a  lower  grade  of  symmetry  than  are  the  holohedrons.  In 
some  cases  only  half  of  the  whole  number  of  planes  found  in  the 
holohedron  occur  on  the  partial  form.  This  condition  is  known 
in  some  instances  as  hemihedrism,  in  others  as  hemimorphism. 
In  other  cases  only  1/4  of  the  planes  persist.  Such  a  condition  is 
known  as  tetartohedrism. 

HOLOHEDRAL  DIVISION. 

(Hexoctahedral  Class.) 

Complete  Symmetry  of  the  System. — The  holohedral  forms 
of  the  isometric  system  possess  three  principal  planes  of  symmetry 
at  right  angles  to  one  another  and  six  secondary  planes  which 
bisect  the  right  angles  between  the  principal  planes  (Fig.  20). 
In  addition  to  these  9  planes  of  symmetry  the  holohedrons  of  this 
system  possess  also  3  principal  axes  of  fourfold  symmetry  that 
are  perpendicular  to  the  three  principal  planes  of  symmetry,  4 
axes  of  threefold  symmetry  passing  through  the  centers  of  the 
octants  included  between  the  three  principal  planes  of  symmetry, 
6  similar  axes  of  twofold  or  binary  symmetry  perpendicular  to  the 
secondary  planes  of  symmetry,  and  one  center  of  symmetry. 


32 


GEOMETRICAL   CRYSTALLOGRAPHY 


The  presence  of  the  center  of  symmetry  demands  the  presence  on 
the  forms  of  pairs  of  parallel  planes  on  opposite  sides  of  the 
center  of  symmetry,  which  must  be  at  the  geometrical  center  of 
the  form. 

Figure  21  represents  in  a  schematic  way  the  elements  of 
symmetry  in  the  holohedrons  of  this  system.  In  this  and  in 
corresponding  figures  on  later  pages  the  characters  of  the  axes 
of  symmetry  are  represented  by  symbols  at  their  terminations 
as  follows:  •  fourfold  or  tetragonal  symmetry;  A  threefold 


FIG.   20. — The  planes  of  symmetry 
of  the  isometric  holohedrons. 


FIG.  21.  — Elements  of 
symmetry  in  the  isometric 
holohedrons. 


or  trigonal  symmetry;  •  twofold  or  binary  symmetry.  Hex- 
agonal symmetry  is  represented  by  •.  The  positions  of  the  prin- 
cipal planes  of  symmetry  are  indicated  by  heavy  lines,  and  those 
of  secondary  planes  by  lighter  lines. 

Axes. — The  lines  chosen  as  the  axes  in  this  system  are  the 
intersections  of  the  three  principal  planes  of  symmetry.  From 
the  relative  positions  of  these  planes  with  respect  to  one  another 
the  axes  must  necessarily  be  three  lines  perpendicular  to  each  other 
at  a  common  point.  Since,  moreover,  a  secondary  plane  of 
symmetry  bisects  each  of  the  angles  between  the  axes,  all  these 
axes  must  be  equivalent;  i.e.,  the  axes  must  be  equal  in  length  (or 
the  distances  taken  as  unity  on  each  must  be  equal)  and  they 
must  be  equally  terminated  by  crystal  planes  (i.e.,  by  the 
same  number  of  similar  crystal  planes). 

The  axes  of  the  isometric  system  may  therefore  be  defined 


THE    ISOMETRIC,    OR   REGULAR,    SYSTEM 


33 


as  three  lines  of  equal  unit  lengths  perpendicular  to  each  other  at 
a  common  point  (Fig.  13,  page  17).  Since  the  axes  are  imagin- 
ary lines  drawn  within  a  crystal,  they  extend  indefinitely. 

It  must  be  borne  in  mind  that  whenever  length  is  referred 
to  in  discussions  of  crystals  relative,  and  not  absolute,  length  is 
meant.  From  the  term  "equal  unit  lengths"  we  must  not  infer 
that  the  unit  length  in  all  crystals  of  the  isometric  system  is  the 
same  absolute  length,  but  rather  that  the  ratio  between  the 
unities  on  the  axes  of  all  isometric  crystals  is  always  one. 

The  Groundform  of  the  System. — The  groundform  of  a 
system  has  been  defined  as  that  form  made  up  of  planes  cutting  the 
axes  at  unity  distances.  In  the  isometric 
system  the  planes  of  the  groundform  cut 
the  three  axes  at  the  same  distance  from 
the  center;  i.e.,  the  ratio  between  their  in- 
tercepts is  i  :  i  :  i.  When  one  plane  of 
this  form  is  present,  symmetry  demands  the 
presence  of  seven  others,  so  that  the  entire 
form  consists  of  eight  planes,  which  together 
comprise  an  octahedron  (Fig.  2  2) .  Its  inter- 
facial  angles  are  each  about  109°  28  1/4'. 

The  parameters  on  the  three  axes  are  unity.  The  symbol  of  a 
plane  is  a  :  b  :  c,  with  the  proper  signs  appended,  or,  since  the 
axes  are  equivalent,  a  :  a  :  a,  and  the  symbol  of  the  form  is  O. 
The  indices  are  i  i  i. 

The  Most  General  Form  in  the  System. — The  most  general 
form  in  the  system  is  that  composed  of  planes  cutting  the  three 
axes  at  different  distances,  no  one  of  which  is  zero  or  infinity. 
Since  one  of  these  distances  may  be  considered  as  unity,  the  sym- 
bol of  one  of  its  planes  is  na  :  b  :  me.  Since,  moreover,  the 
different  axes  are  equivalent  or  since  the  symmetry  of  the  sys- 
tem requires  that  each  axis  shall  be  terminated  in  the  same 
manner,  it  follows  that  the  presence  of  the  plane  na  :  b  :  me 
necessitates  the  presence  of  five  other  planes  in  the  same  octant, 
for  without  introducing  any  other  than  the  plus  signs  understood  in 
the  above  symbol,  this  symbol  may  be  changed  five  times  by 
3 


FIG.  22. — Groundform  of 
the  isometric  system. 


34  GEOMETRICAL   CRYSTALLOGRAPHY 

using  the  same  parameters  before  each  of  the  three  axes  success- 
ively, as: 

na  :    b  :  me  na  :  mb  :  c  a  :  mb  :  nc 

a  :  nb  :  me  ma  :  nb  :  c  ma  :     b  :  nc 

There  are  thus  in  this  form  six  planes  in  an  octant.  As  there 
are  eight  octants  in  all,  the  total  number  of  planes  in  the  form 
must  be  48. 

We  can  now  understand  why  in  this  system  we  may  write  the 
symbol  of  the  axes  a  :  a  :  a  instead  of  a  :  b  :  c.  Each  axis  is 
affected  in  the  same  manner  by  similar  planes.  There  is  no 
character  by  which  one  axis  is  distinguished  from  the  others,  and 
hence  we  may  indicate  them  all  by  the  same  letter.  The  symbol 
of  the  individual  planes  is  na  :  a  :  ma,  with  the  proper  signs,  and 
that  of  the  form  n  O  m  or  m  O  n.  Its  indices  are  h  k  I,  in  which 
the  three  different  letters  stand  for  three  different  values. 

The  Hexoctahedron. — The  most  general  form  of  the  system 
is  known  as  the  hexoctahedron  (Fig.  23),  because  it  possesses  six 
faces  in  an  octant;  or  it  is  a  form  that  may  be  regarded  as  an 
octahedron  in  which  an  octahedral  face  is 
replaced  by  six  equal  faces,  each  of  which  is 
a  triangle.  Each  face  cuts  one  axis  at  a 
distance  taken  as  unity,  another  at  n  times 
this  distance,  and  the  third  at  m  times  this- 
distance.  The  interfacial  angles  are  of  three 
FIG.  23.— Hexoctahe-  kinds  (o,  c,  and  d  in  Fig.  23).  Their  values 

dron,  mOn,  or  hkl.        yarv   Wjtj1   ^   va]ues    of   m  and  Hj   the   figure 

approaching  more  and  more  nearly  the  habit  of  the  octahedron 
as  m  and  n  approach  unity  in  value.  The  symbols  203  and 
364  both  represent  hexoctahedra.  They  signify  forms  made 
up  of  planes  cutting  the  three  axes  at  distances  whose  ratios  are 
2:1:3  and  3:1:4.  Their  indices  are  3  6  2  and  4  12  3. 

Other  Forms  Derived  from  the  General  Form. — In  the 
symbol  for  the  general  form  na  :  a  :  ma,  the  parameters  n  and  m 
may  be  made  to  vary  between  o  and  <x> .  If  we  give  n  and  m  all  the 
different  values  possible,  remembering  that  a  plane  which  does 
not  meet  the  axis  cuts  it  at  o° ,  we  obtain  the  following  symbols : 


THE   ISOMETRIC,    OR   REGULAR,    SYSTEM  35 

ma  :  a  :  ma,  when  n  —  m;  i.e.,  when  both  parameters  are  equal, 

and  neither  is  unity  nor  infinity. 
a  :  a  :  ma,  or  na  :  a  :  a,  when  m  or  n=  i ;  i.e.,  when  one  of  these 

parameters  becomes  unity. 
GO<Z  :  a  :  ma,  or  na  :  a  :  o>a,  when  m  or  n=  oo;  i.e.,  when  one  of 

these  parameters  becomes  infinity. 
a  :  a  :  a,  when  m  and  w=i;  i.e.,  when  both  parameters  become 

unity. 

oca  :  a  :  GO  a,  when  m  and  n=  °o ;  i.e.,  when  both  parameters  be- 
come infinity. 
a  :  a  :  000,  or  co#  :  a  :  a,  when  m  orn=i,  and  the  remaining n  or 

w  equals  infinity. 

Thus  there  are  only  seven  possible  kinds  of  forms  in  the  regu- 
lar system,  for  a  :  a  :  ma  and  na  :  a  :  a  are  the  same  kind  of  form, 
as  are  also  ooa  :  a  :  ma  and  na  :  a  :  <xa,  and  a  :  a  :  &a  and 
GO#  :  a  :  a. 

The  last  two  symbols  are  symbols  of  different  planes  on  the 
same  form,  while  coa  :  a  :  ma  and  na  :  a  :  GO  a  are  forms  in  which 
the  parameter  which  is  neither  unity  nor  infinity  is  some  other 
value  which  is  different  in  the  two  forms.  One  may  be  2  O  GO  and 
the  other  3  O  GO,  on  each  of  which  the  distribution  of  the  planes  is 
the  same  although  their  interfacial  angles  are  different. 

Forms  Composed  of  24  Planes. — Of  the  forms  indicated 
above  three  possess  24  faces  each.  These  are  those  made  up  of 
planes  with  the  symbols  ma  :  a  :  ma,  a  :  a  :  ma,  and  GO  a  :  a  :  ma 
respectively.  The  symbols  of  the  forms  are  m  O  m,  m  O,  and 
oo  O  m.  The  values  of  their  interfacial  angles  depend  upon  the 
value  of  m.  The  corresponding  indices  are  Ml,  hlh  and  hlo  in 
which  h>l. 

The  Icositetrahedron. — The  first  of  these  is  the  icositetrahedron 
(Fig.  24)  or  the  24-sided  figure,  consisting  of  a  polyhedron  bounded 
by  24  similar  trapeziums,  three  in  each  octant.  Each  plane  cuts 
one  axis  at  unity  and  the  other  two  axes  at  some  distance  greater 
than  unity  and  less  than  infinity.  The  interfacial  angles  are  two 
in  kind  (o  and  c  of  Fig.  23).  Examples  202  and  3  O  3,  or  211 
and  311. 

The  Trisoctahedron. — The  second  form,  known  as  the  trisoc- 


GEOMETRICAL    CRYSTALLOGRAPHY 


tahedron  (Fig.  25)  is  bounded  by  three  isosceles  triangles  in  each 
octant,  so  distributed  that  their  apices  meet  at  a  common  point, 
which  is  equally  distant  from  the  terminations  of  all  the  axes. 
Each  plane  cuts  two  axes  at  unity  and  one  at  m  and  each  makes 
with  the  contiguous  planes  two  different  interfacial  angles  o  and  d 
(of  Fig.  23).  Examples  2  O  and  3  O,  or  212  and  313. 


FIG.  24. — Icositetrahe- 
dron,  m  O  m  or  ML 


FIG.  25. — Trisoctahe- 
dron,  mO,  or  hlh. 


The  Tetrahexahedron. — The  remaining  form  is  also  bounded 
by  24  isosceles  triangles  (Fig.  26),  but  they  are  in  groups  of  four 
with  their  apices  at  a  common  point  which  is  the  termination  of 
an  axis.  Each  plane  meets  one  axis  at  unity,  another  at  some 
distance  greater  than  unity,  and  is  parallel  to  the  third.  Two 
different  interfacial  angles  are  present.  They  correspond  to  the 
c  and  d  edges  of  the  hexoctahedron.  Examples  o>  O  2,  oo  04, 
3  O  oo;  or  210,  410,  and  310. 


FIG.   26. — Tetrahexahe- 
dron, wOoo  orhl°. 


FIG.   27. — Dodecahe- 
dron, oo  O  or  no. 


Form  Composed  of  12  Planes. — The  Dodecahedron  possesses 
12  planes  (Fig.  27),  each  of  which  is  at  the  same  time  in  two 
octants.  All  are  similar  rhombs,  four  of  which  meet  at  each 
termination  of  the  axes,  and  each  cuts  two  axes  at  the  same  dis- 
tance and  is  parallel  to  the  third.  Their  symbols  are  a  :  a  :  ooa. 


THE   ISOMETRIC,    OR   REGULAR,    SYSTEM 


37 


The  interfacial  angles  are  each  120°,  and  since  there  are  no 
variable  parameters  possible  in  the  form  there  can  be  only  one 
dodecahedron  in  the  system.  The  symbol  of  the  form  is  ooQ 
and  its  indices  no. 

Form  Composed  of  8  Planes. — The  Octahedron,  O,  has 
already  been  discussed.  It  consists  of  eight  equilateral  triangular 
faces,  each  in  an  octant.  Its  interfacial  angles  measure  109° 
28'  16".  (See  Fig.  28.) 

Form  Composed  of  6  Planes. — The  Cube,  ooO^,  contains 
the  smallest  number  of  planes  possible  to  holohedrons  in  the 
isometric  system  (Fig.  29).  Each  of  the  planes  is  a  square,  one 
of  which  is  perpendicular  to  each  axis  at  its  termination.  The 
interfacial  angles  are  all  90°.  The  indices  are  100. 


FIG.  28. — Octahedron,  O  or  in. 


FIG.  29. — Cube, oo  Oooor  100. 


Summary. — The  total  number  of  holohedral  forms  possible  in 
the  isometric  system  is  seven.  Of  these,  one  possesses  48  planes, 
3  possess  24  planes  each,  one  possesses  12  planes,  one,  8  planes, 
and  one  6  planes.  Of  these  three,  the  octahedron,  the  cube,  and 
the  dodecahedron,  are  forms  that  possess  constant  interfacial 
angles,  consequently  only  one  of  each  is  possible.  The  symbols 
of  the  other  four  forms  possess  variable  parameters,  hence  there 
may  be  as  many  of  each  of  these  forms  in  the  system  as  there 
are  values  that  may  be  assumed  by  their  parameters.  Their 
number  is  limited  only  by  the  law  of  the  rationality  of  the  indices. 

Determination  of  the  Forms.— On  crystals  the  symbols  that 
represent  the  different  forms  are  determined  from  the  interfacial 
angles  made  by  their  planes  with  each  other.  By  measurement 
with  the  goniometer  the  values  of  the  interfacial  angles  between 
selected  planes  are  obtained.  From  these,  by  the  methods  of 


GEOMETRICAL    CRYSTALLOGRAPHY 


spherical  geometry,  the  value  of  the  intercept  on  one  axis  may  be 
calculated  in  terms  of  its  intercept  on  another.  If  the  latter  be 
taken  as  the  unity  intercept  the  value  determined  is  the  parameter 
of  the  plane  on  the  other  axis. 

If  in  the  figure  (o>Ow,  Fig.  30)  the  angle  C  is  measured,  then 
by  calculation  we  can  determine  the  relative  distances  at  which 
the  plane  t  cuts  the  two  axes  Oa  and  Oc.  If  the  distance  at 

which  the  plane  cuts  Oa  be  taken 
as  unity,  then  the  intercept  on  Oc 
measured  in  terms  of  this  unity 
is  the  parameter  on  Oc.  If  the 
value  of  the  angle  is  126°  52', 
calculation  will  show  that  the 
plane  cuts  Oc  at  3  times  the  dis- 
tance at  which  it  cuts  Oa.  Its 
symbol  is  a  :  oca  :  3^  or  3  O  a>. 
Simple  Forms  and  Combi- 
nations.— Each  of  the  figures 
reproduced  above  represents  a 
simple  form;  i.e.,  a  polyhedron 
all  of  whose  planes  are  necessi- 
tated by  the  presence  of  a  single  one.  All  the  planes  on  each 
form  may  be  represented  by  a  single  symbol. 

In  nature,  while  crystals  occur  that  are  bounded  by  the 
planes  of  a  single  form,  it  is  more  usual  to  find  on  them  planes 
belonging  to  several  forms.  Such  crystals  must  be  represented  by 
as  many  symbols  as  there  are  planes  on  them  belonging  to  different 
forms.  The  total  number  of  planes  present  is  equal  to  the  sum 
of  all  the  planes  belonging  to  all  the  forms  represented  by  the 
symbols. 

If  a  crystal  contains  planes  belonging  to  the  forms  203,  co  O 
and  ooOoo,  then  the  total  number  of  planes  on  the  crystal  must 
be  48+12  +6  or  66. 

The  occurrence  of  two  or  more  forms  on  a  crystal  is  known  as  a 
combination  of  forms.  The  symbol  of  a  combination  consists 
of  the  symbols  of  its  different  forms  written  in  order,  with  the 
symbol  for  the  form  with  the  largest  faces  first. 


THE   ISOMETRIC,    OR   REGULAR,    SYSTEM  39 

The  forms  occurring  on  the  crystal  represented  in  figure  31 
are  the  cube  (h)  and  the  dodecahedron  (d).  Its  symbol  is 
ooQoo,  oo  O.  In  figure  32  we  have  combinations  of  the  cube, 
octahedron,  and  dodecahedron.  On  the  crystal  to  the  left 
O  predominates,  on  that  to  the  right  oo  O  oo  is  the  largest  form. 
The  symbols  for  the  combinations  are  O,  ooQ  oo,  ooO  and  ooO  oo, 
O,  oo  O.  When  written  in  the  proper  order 
they  indicate  the  general  habit  of  the  crystal 
which  they  represent.  The  first  of  these 
two  symbols  indicates  that  the  crystal  which 
it  represents  is  octahedral  in  habit,  and 
the  second  that  the  corresponding  crystal 

possesses  a  Cubical  habit.  FlG    3I. -Combination 

NOTE. — In  writing  the  symbols  of  com-  of  cube  (ooOco)and  do- 

.  .        .          .     .  r  /-       i  •      decahedron  (ooO). 

binations  it  is  necessary  first  to  fix  the  posi- 
tion of  the  axes.  This  is  done  by  determining  the  positions  of 
the  principal  planes  of  symmetry  in  the  crystal  (or  model)  and 
noting  their  lines  of  intersection.  These  three  lines  of  intersec- 
tion are  the  three  axes  to  which  all  the  planes  must  be  referred. 
When  once  fixed  they  remain  the  axes  for  all  the  planes  in  the 
combination.  If  the  choice  of  the  axes  is  the  correct  one,  their 
six  ends  terminate  at  similar  points  on  the  crystal. 


FIG.  32. — Combinations  of  octahedron,  cube  and  dodecahedron. 

The  next  step  is  to  determine  the  relations  of  the  several 
faces  to  these  axes.  A  close  inspection  of  the  planes  will  often 
suffice  to  indicate  the  approximate  relative  distances  at  which 
they  cut  the  axes.  If  this  is  not  easily  seen,  a  glass  plate  laid  flat 


40  GEOMETRICAL    CRYSTALLOGRAPHY 

against  a  plane  will  serve  to  aid  in  this  determination.  The 
distances  from  the  center  of  the  crystal  at  which  the  plate  cuts 
the  axes,  or  their  prolongations,  afford  the  means  for  estimating 
the  lengths  of  the  parameters  of  the  plane  on  which  the  plate  lies. 
Suppose,  for  instance,  the  glass  plate  has  been  laid  on  a  plane 
whose  symbol  is  sought.  If  the  plate  meets  one  axis  at  a  certain 
distance,  a  second  at  some  greater  distance,  and  the  third  at  a  still 
greater  distance,  then,  since  the  shortest  of  these  distances  may  be 
made  unity,  the  symbol  of  the  plane  is  a  :  ma  :  na  or  the  form  to 
which  it  belongs  is  m  O  n.  Now,  suppose  the  plate  be  laid  on  a 
second  plane.  It  may  cut  one  axis  at  a  certain  distance  and 
the  other  two  at  similar  distances  which,  however,  are  different 
from  the  distance  at  which  it  cuts  the  first  axis.  The  symbol  of 
the  plane  is  either  a  :  ma  :  ma  ,  or  ma  :  a  :  a.  The  shortest 
distance  is  always  made  unity.  Consequently,  if  the  two  axes 
that  are  cut  at  the  same  distance  are  cut  nearer  the  center  of  the 
crystal  than  is  the  other  axis,  the  symbol  of  the  plane  is  ma  :  a  :  a, 
which  is  a  plane  of  form  mO.  If,  on  the  other  hand,  these  two 
axes  are  cut  at  a  greater  distance  from  the  center  than  the  third 
axis,  the  plane  is  a  :  ma  :  ma,  or  a  plane  of  the  form  m  O  m. 


CHAPTER  VI. 

PARTIAL  FORMS — HEMIHEDRISM  AND  TETARTOHEDRISM 
OF  THE  ISOMETRIC  SYSTEM. 

Independent  Occurrence  of  Partial  Crystal  Forms. — The 

holohedral  forms  that  have  been  discussed  are  those  composed  of 
the  full  number  of  planes  which  the  complete  symmetry  of  the  iso- 
metric system  demands.  Sometimes,  in  consequence  of  irregular 
growths,  certain  planes  on  a  crystal  may  be  developed  at  the  ex- 
pense of  other  planes,  until  finally  in  extreme  cases  the  latter  may 
be- reduced  to  mere  points,  and  so  disappear  as  planes.  This  ir- 
regular disappearance  of  some  planes  of  a  complete  form  is  acci- 
dental, and  is,  consequently,  of  no  crystallographic  significance. 
The  condition  is  known  as  merohedrism,  and  the  incomplete  form 
is  said  to  be  merohedral. 

In  the  crystal  of  quartz  (SiO2)  represented  in  figure  u  the 
growth  was  not  symmetrical  about  the  axis,  and  so  the  crystal 
is  distorted.  The  plane  z  at  the  upper  left-hand  corner  has  been 
crowded  by  the  abnormal  development  of  the  planes  P  and  r,  until 
it  has  nearly  disappeared.  If  the  crystal  had  continued  its  growth 
in  the  same  direction,  z  would  probably  have  disappeared,  and 
the  form  to  which  it  belongs  would  have  been  represented  by  one 
less  plane  than  normally  belongs  to  it. 

Hemimorphism.— Occasionally  all  the  planes  of  half  the 
holohedral  form  occur  at  one  termination  of  an  axis  of  symmetry, 
while  from  the  other  end  of  the  same  axis  they  are  absent,  or  are 
represented  by  all  the  planes  belonging  to  half  of  some  other  form; 
i.e.,  the  symmetry  axis  is  polar.  It  is  to  be  noted  that  in  cases  like 
this  all  the  planes  of  the  given  holohedral  form  that  are  naturally 
expected  to  be  present  around  one  termination  of  an  axis  are 
present  while  none  are  present  at  the  other  end  of  the  same  axis. 
This  condition  is  known  as  hemimorphism.  It  is  characteristic 
of  a  few  minerals  which  always  exhibit  it  and  hence  must  be  the 
direct  result  of  the  internal  arrangement  of  their  molecules. 


42  GEOMETRICAL   CRYSTALLOGRAPHY 

That  this  is  the  case  is  shown  by  the  physical  properties  of 
hemimorphic  crystals.  When  these  are  heated  they  become  pyro- 
electric,  one  end  becoming  charged  with  positive  electricity  and 
the  other  end  with  negative  electricity.  The  former  is  known  as 
the  analogue  pole,  and  the  latter  as  the  antilogue  pole.  In  writing 
the  symbols  of  hemimorphic  crystals  the  pole  about  which  the 
several  hemimorphic  forms  occur  must  be  indicated.  (See  also 
pages  216-217.) 

Figure  33  represents  a  hemimorphic  crystal  of  calamine 
(Zn(OH)2SiO3),  an  orthorhombic  mineral.  The  two  ends  of 
the  c  axis  are  differently  terminated,  the 
lower  end  being  terminated  by  one-half  the 
planes  of  one  form  and  the  upper  end  by 
one-half  the  planes  belonging  to  two  other 
forms. 

Hemihedrism   and  Tetartohedrism.— 
In   addition    to   the  merohedral  and  hemi- 
morphic forms  there  are  other  partial  forms 
FIG.  33.— Crystal  exhib-  observed   on   crystals  that  are  of  great  im- 

iting  hemimorphism.  ,     .  , 

portance,  since  their  presence  characterizes  a 
large  number  of  substances.  These  forms  do  not  possess  the 
symmetry  of  the  holohedrons  belonging  to  any  system,  but  they 
are  bounded  by  planes  that  are  the  same  geometrically  as  those 
on  certain  holohedrons  from  which  they  may  be  regarded  as 
being  derived.  When  the  planes  that  persist  in  the  partial  form 
are  one-half  of  those  present  on  the  corresponding  holohedron, 
the  new  form  is  called  a  hemihedron,  or  it  is  said  to  be  hemihedral. 
When  only  one-quarter  of  the  planes  persist,  the  new  form  is 
tetartohedral. 

Law  of  Hemihedrism  and  Tetartohedrism. — If  the  only 
condition  of  hemihedrism  and  tetartohedrism  were  the  occurrence 
of  one-half  or  one-quarter  of  the  planes  of  the  holohedron,  the 
number  of  new  hemihedrons  and  tetartohedrons  that  might  exist 
would  be  very  large.  Experience  shows,  however,  that  only  those 
planes  persist  in  the  partial  forms  which  terminate  equivalent  ends 
of  the  crystallographic  axes  in  a  similar  manner.  In  other  words, 
in  hemihedrons  and  tetartohedrons  which  are  not  hemimorphic, 


HEMIHEDRISM   OF    THE   ISOMETRIC    SYSTEM  43 

equivalent  terminations  of  the  crystallo graphic  axes  are  equivalently 
terminated.  That  is:  All  equivalent  terminations  of  the  axes  in 
hemihedrons  and  tetartohedrons  (terminations  that  are  separated 
by  planes  of  symmetry  in  the  holohedrons)  must  be  cut  by  the 
same  number  of  similar  planes.  This  limitation  of  the  planes 
that  occur  in  the  new  forms  reduces  their  number  to  compara- 
tively few. 

We  may  imagine  any  hemihedral  or  tetartohedral  form  as 
being  produced  by  the  suppression  of  a  certain  half  or  three- 
quarters  of  the  planes  composing  the  complete  or  holohedral 
form  and  the  extension  of  the  remaining 
planes  until  they  meet.  Suppose,  for  in- 
stance, that  the  white  planes  of  the  inside 
figure  (Fig.  34)  are  suppressed  and  that 
the  shaded  planes  are  extended  to  intersec- 
tion: they  will  then  produce  the  outside 
figure,  which  is  the  hemihedron  corre- 
sponding to  the  interior  holohedron.  In 
this  crystal  the  terminations  lettered  A,  B, 
C,  D  are  all  equivalent  in  the  holohedral 

r  i      •       i       i          -i      i      T   p  FIG.  34. — Crystal  exhibit- 

form,  consequently  in  the  hemihedral  form  ing  hemihedrism.   Outside 

they  must  be  cut  by  the  same  number  f°rm  contains  one-half  the 

J  planes  of  the  enclosed  form. 

of  equivalent  planes.      An  inspection  of 

the  figure  will  show  that  each  is  cut  by  two  planes  which 
are  equally  inclined  to  each  axis.  The  axis  from  E  to  F  is 
not  equivalent  to  the  axes  terminating  at  A,  B,C,D,  hence  its 
extremities  are  not  terminated  in  the  same  way  as  are  A,B,C,D. 
But  there  is  a  plane  of  symmetry  in  the  holohedral  form  perpen- 
dicular to  this  axis,  hence  its  two  terminations  are  equivalent, 
and  consequently  in  the  hemihedron  they  should  be  similarly 
terminated.  As  a  matter  of  fact,  E  and  F  terminate  in  three 
planes  each,  and  the  planes  that  cut  at  E  are  inclined  to  the  axis 
at  exactly  the  same  angles  as  those  at  which  the  three  planes 
cutting  at  F  are  inclined  to  it. 

Combinations  of  Hemihedral  and  Tetartohedral  Forms.— 
As  in  the  case  of  holohedral  forms,  combinations  of  hemihedral 
and  of  tetartohedral  forms  are  frequently  met  with  in  nature. 


44  GEOMETRICAL   CRYSTALLOGRAPHY 

Since,  however,  the  external  forms  of  crystals  are  but  the  expres- 
sions of  definite  plans  of  internal  structure,  it  must  necessarily 
follow  that  combinations  of  forms  are  limited  to  those  of  the 
same  grade  of  symmetry.  (See  statement  of  Law  of  Symmetry, 
page  28.)  Holohedral  forms  combine  with  holohedral  forms; 
hemihedral  with  hemihedral,  and  tetartohedral  with  tetartohedral. 
Further,  only  those  hemihedral  forms  possessing  the  same 
symmetry  may  combine  with  one  another,  and  only  those  tetar- 
tohedrons  with  the  same  symmetry. 

Some  holohedral  forms,  when  subjected  to  the  suppression  of 
one-half  or  of  three-quarters  of  their  planes  yield  by  the  extension 
of  the  remaining  planes  new  forms  that  are  geometrically  identical 
with  the  originals;  i.e.,  they  are  hemihedrons  that  are  identical 
geometrically  with  holohedrons.  Such  apparently  holohedral 
forms  may  combine  with  hemihedral  or  with  tetartohedral  forms 
that  may  be  considered  as  having  been  derived  from  other 
holohedral  forms  by  the  same  method  as  was  applied  to  the 
forms  that  appear  to  be  holohedrons,  and  thus  there  is  seemingly 
an  exception  to  the  general  statement  made  above.  But  this 
supposed  exception  is  not  a  real  one,  since  the  apparently  holo- 
hedral forms  are  found  to  possess  all  the  properties  of  hemihedral 
ones,  except  that  of  shape. 

HEMIHEDRISM  OF  THE  ISOMETRIC  SYSTEM. 

Number  of  New  Hemihedral  Forms  in  the  Isometric 
System. — The  whole  number  of  new  hemihedral  forms  that  are 
directly  related  to  the  holohedrons  of  the  isometric  system  is 
seven.  The  other  hemihedral  forms  in  this  system  are  geometric- 
ally identical  with  holohedrons. 

Grouping  of  the  Hemihedrons. — The  seven  new  hemi- 
hedrons of  the  isometric  system  may  be  grouped  into  three  classes 
in  accordance  with  their  grade  of  symmetry.  They  are  all  of  a 
lower  grade  of  symmetry  than  the  holohedrons  of  the  system,  as 
would  be  expected  from  the  fact  that  they  possess  fewer  faces 
than  those  demanded  by  the  condition  of  holohedrism.  Some  of 
them  have  lost  all  the  principal  planes  of  symmetry  that  are 
found  in  the  holohedrons,  others  have  lost  all  the  secondary 


HEMIHEDRISM    OF    THE   ISOMETRIC    SYSTEM  45 

planes  of  symmetry,  and  another  has  lost  all  of  its  symmetry 
planes  of  both  kinds. 

The  forms  of  each  of  the  three  groups  may  be  considered  as 
being  derived  from  holohedrons  by  suppressing  a  certain  half  of 
the  holohedral  planes  and  allowing  the  other  half  to  extend  until 
they  intersect.  In  order,  however,  that  the  persisting  planes  may 
comply  with  the  condition  of  hemihedrism,  which  demands  that 
in  the  hemihedron  equivalent  terminations  of  the  crystallographic 
axes  must  be  equivalently  terminated,  the  derivation  of  the 
hemihedrons  from  holohedrons  must  take  place  in  one  of  three 
ways,  each  of  which  gives  rise  to  one  of  the  groups  already 


FIG.  35.  FIG.  36.  FIG.  37. 

Figures  illustrating  the  three  possible  methods  of  derivation  of  the  hemihedrons 
in  the  isometric  system.  Fig.  35,  gyroidal,  Fig.  36,  pentagonal,  and  Fig.  37  tetra- 
hedral. 

referred  to.     The  only  three  possible  ways  by  which  the  desired 
result  may  be  accomplished  are  illustrated  in  the  three  figures 

(Figs-  35>  36,  37)- 

The  first  figure  represents  the  manner  in  which  hemihedrons 
without  planes  of  symmetry  may  be  derived;  i.e.,  by  the  suppres- 
sion of  alternate  planes  on  the  hexoctahedron.  Figure  36  repre- 
sents the  way  in  which  hemihedrons  retaining  the  principal  planes 
of  symmetry  may  be  derived;  i.e.,  by  the  suppression  of  pairs  of 
hexoctahedral  planes  that  intersect  in  the  principal  planes  of 
symmetry,  or  those  holohedral  planes  that  lie  in  the  alternate 
sections  of  the  12  included  within  the  secondary  planes  of  sym- 
metry occurring  in  the  holohedrons.  Figure  37  represents  the 
way  in  which  the  third  group  of  hemihedrons  may  be  derived; 
i.e.,  by  the  suppression  of  all  the  planes  in  alternate  octants. 

If  we  imagine  either  the  white  or  the  shaded  planes  to  disap- 


46  GEOMETRICAL   CRYSTALLOGRAPHY 

pear  and  the  others  to  be  extended  until  they  intersect,  three  new 
forms  will  result,  each  of  which  will  satisfy  the  conditions  of 
hemihedrism.  No  other  method  of  suppressing  half  the  planes 
on  this  form  will  result  in  new  forms  complying  with  these  con- 
ditions. Thus  three  distinct  kinds  of  hemihedrism  are  possible 
in  this  system,  and  only  three.  The  new  forms  are  known  as 
gyroidal,  parallel,  or  pentagonal,  and  inclined  or  tetrahedral. 

Gyroidal  Hemihedrism  (Pentagonal  Icositetrahedral  Class).— 
Although  gyroidal  hemihedral  forms  are  known  on  crystals,  they 
are  rare  and  consequently  they  will  not  be  discussed.  One  of  the 


FIG.    38. — Right    pentagonal  FIG.    39. — Left  pentagonal 

icositetrahedron,  r  ^°J^'  icositetrahedron,  /  m°n" 

two  gyroidal  hemihedrons  derived  from  mOn  is  represented  in 
figure    38.      This    i,s    known    as    the    right  form    and    has  the 

symbol  r-      ,  or  r  (hkl).     The  left  form,  I—  — ,   or   r  (M),  is 
2  2 

represented  in  figure  39. 

Pentagonal  Hemihedrism  (Dyakisdodecahedral  Class).— 
The  different  types  of  pentagonal  hemihedrons  are  two  in 
number.  One  is  derived  from  the  holohedron  mOn  and  the 
other  from  mO  oo .  Both  retain  the  three  principal  planes  of 
symmetry,  the  four  trigonal  axes  of  symmetry  and  the  center  of 
symmetry.  The  three  axes  of  fourfold  symmetry  become  axes 
of  binary  symmetry.  These  elements  are  indicated  in  figure  40, 
which  indicates  the  character  of  the  symmetry  that  would  result 
if  all  the  shaded  planes  of  mOn  or  all  the  white  ones  should 
disappear.  In  other  words,  it  represents  the  symmetry  of  all 
the  white  planes  alone  or  of  all  the  shaded  ones.  The  heavy 


HEMIHEDRISM   OF   THE   ISOMETRIC   SYSTEM 


47 


solid  lines  indicate  the  positions  of  the  planes  of  symmetry  that 
would  remain,  and  dotted  lines  the  positions  of  those  that  would 
drop  out. 

The  Diploids. — By  the  application  of  the  pentagonal  method 
of  selection  to  the  planes  of  the  hexoctahedron,  two  hemihedral 
forms  are  produced  that  differ  according 
to  the  positions  of  the  original  planes 
that  survive.  (Figs.  41  and  42.)  Both 
are  known  as  diploids.  They  are  iden- 
tical in  shape,  but  differ  in  their  posi- 
tions with  respect  to  the  axes.  Either 
one,  by  the  revolution  of  90°  about 
either  of  its  crystallographic  axes,  may 
be  brought  exactly  into  the  position  of  FIG.  40.— Diagram  illustrating 

,,        ,1  -r,  ,1,1  ,i  •        i    -•         the  distribution  of  the  elements 

the  other.     Forms  that  bear  this  relation  of    symmetry  in   pentagonal 
to  each  other  are  said  to  be  congruent,  hemihedrons. 
The  one  outlined  in  heavy  lines  in  figure  41  is  designated  as  the 
positive  dyakisdodecahedron,  or  diploid,  and  the  other  (Fig.  42)  as 

fwO^I  [mOn  1 

and-         -. 
L    2    J  L    2     J 

Their  indices  are  n(hkl)  and  nfylk).     Each  form  is  bounded  by 


the   negative   one.     Their  symbols  are 


FIG.  41. — Positive  diploid 
or  dyakisdodecahedron, 


FIG.  42. — Negative  diploid 
or  dyakisdodecahedron, 
" 


twenty-four  similar  trapeziums  meeting  in  three  kinds  of  inter- 
facial  angles.  There  are,  of  course,  as  many  pairs  of  diploids 
possible  as  there  are  hexoctahedra.  Their  shapes  will  necessarily 
vary  with  the  values  of  m  and  n. 

The  Pentagonal   Dodecahedrons   or   the  Pyritohedrons. — The 
only  other  hemihedron  of  this  class  may  be  derived  from  the 


4o  GEOMETRICAL   CRYSTALLOGRAPHY 

tetrahexahedron.  The  extension  of  alternate  planes  on  the 
holohedron  corresponds  exactly  to  the  extension  of  alternate  pairs 
of  hexoctahedral  planes  that  intersect  in  the  planes  of  symmetry. 
The  new  forms  derived  by  the  extension  of  alternate  planes 
of  the  tetrahexahedron  are  the  pyritohedrons  (Figs.  43  and  44). 


FIG.  43. — Positive  py- 
ritohedron,  +  j^M00'J  Or 
ir(hlo). 


FIG.  44.— N  e  g  a  t  i  v  e 
pyritohedron,  —[-^2] 
or  v(hcl). 


They  are  congruent  forms  bounded  by  twelve  similar  pentagons 
meeting  in  two  different  interfacial  edges.     Their  symbols  are 

+ 


]FwO  ooH 
(Fig.  43)  and  -  (Fig.  44).     Their  indices  are 

x(hlo)  and  n(hol). 

Tetrahedral    Hemihedrism    (Hextetrahedral    Class).— The 

extension   of   all   the   planes  occurring  in  alternate   octants   of 

holohedral  forms  in  the  isometric  system 
and  the  suppression  of  those  in  the 
other  octants  will  produce  new  forms 
in  all  cases  except  the  tetrahexahedron, 
the  dodecahedron,  and  the  cube.  These 
three  holohedrons  yield  hemihedral 
forms  that  are  geometrically  indis- 
tinguishable from  themselves.  Of  the 

FIG  45  —  Diagram  iiiustra-  four  new  geometrical  forms  thus  pro- 
ting  the  distribution  of  the  ele-  r  . 

ments  of  symmetry  in  tetrahe-  duced    all    have    a    tetrahedral    habit, 

hence    the    name    of    the   class.      The 

forms  are  congruent,  each  holohedron  yielding  two  hemihedrons, 
which  are  distinguished  as  the  positive  and  negative  forms. 

In  the  tetrahedral  hemihedrons  the  principal  planes  of  sym- 


HEMIHEDRISM   OF   THE   ISOMETRIC   SYSTEM 


49 


,metry,  the  six  axes  of  binary  symmetry,  and  the  center  of  sym- 
metry have  disappeared.  The  three  axes  of  fourfold  symmetry 
have  become  axes  of  binary  symmetry.  There  remain  the  six 
secondary  planes  of  symmetry  and  the  four  trigonal  axes  of  sym- 


FIG.  47. — Negative  hex- 
tetrahedron,   —    m°H  or 


Fig.  46. — Positive  hex- 
tetrahedron,    +  -^2-    or 

K(hkl). 


metry.     These  axes  are  now,  however,  polar.     The  forms  con- 
tain no  parallel  planes  (Fig.  45). 

The   Hextetrahedrons. — The    hextetrahedrons    are    24-sided 
figures  bounded  by  scalene  triangles  meeting  in  3  kinds  of  edges. 


FIG.   48. — Positive    tristetrahe- 
dron,  +  -?£*   or  K(hll). 


FIG.    49. — Negative    tristetrahe- 
dron,  —  —  —  or  ic(hlfy. 


The  crystal  axes  terminate  in  the  solid  angles  at  which  4  planes 
meet.     Figure  46  is  the  positive  form,  +-   — ,  and  figure  47,  the 

/~\ 

negative  form,  -       — .     Their  indices  are  *(hkl}  and  K.(hkl). 

The  Trigonal  Tristetrahedrons. — The  trigonal  tristetrahedrons 
are  composed  of  half  the  planes  of  the  icositetrahedron.     Each  is 


5° 


GEOMETRICAL    CRYSTALLOGRAPHY 


bounded  by  12  similar  isosceles  triangles  intersecting  in  two  kinds 
of  edges.     The  crystal  axes  terminate  in  the  centers  of  the  long 

edges.     Figure   48  represents    the  positive    form,    +-     — ,  and 


figure  49,  the  negative  one,  - 
are  *(hll)  and  "(hll). 


mOm 


Their  corresponding  indices 


FIG.  50. — Positive  del- 
toid dodecahedron,  +  — 
or  K(hlh) 


FIG.  51. — Negative 
deltoid      dodecahedron, 

—  mO.    OTK(Mh}. 


The  Tetragonal  Tristetrahedrons  or  Deltoid  Dodecahedrons.— 
There  are  two  of  these  derived  from  the  trisoctahedron  as  indi- 
cated in  figures  50  and  51.  They  are  bounded  by  12  trapeziums 
intersecting  in  two  kinds  of  interfacial  angles.  The  crystallo- 


FIG.  52. — Positive  tetrahedron 
+  —  or  K(III). 


FIG.  53. — Negative  tetrahe- 
dron, — —  or  K(III). 


graphic  axes  terminate  in  the  solid  angles  at  which  four  planes 

meet.     Their  symbols  are   H —  -  (Fig.  50)  and  -  (Fig.  51), 

2  2 

and  their  indices  are  "(hlh)  and  "(hlh). 

The  Tetrahedrons. — These  forms  are  derived  from  the  octa- 
hedron.    There  are  two  of  them,  each  bounded  by  four  equi- 


HEMFHEDRISM    OF    THE    ISOMETRIC    SYSTEM 


51 


lateral  triangles  intersecting  in  six  similar  interfacial  edges  (Figs. 
52  and  53).     The  crystallographic  axes  terminate  in  the  centers  of 

these  edges.     Their  symbols  are   H —  and   -  — ,     Their  indices 

are  *(iii)  and  K(III). 

NOTE. — The  hemihedrons  derived  from  mOm  possess  tri- 
angular faces,  while  the  holohedrons  are  bounded  by  trapeziums. 
On  the  other  hand,  the  hemihedron  derived  from  mO  is  composed 
of  trapeziums,  while  the  faces  of  the  holohedron  are  triangles. 

TABULAR    SUMMARY    OF    ISOMETRIC    HEMIHEDRONS. 

( The  forms  with  symbols  not  written  as  fractions  are  indistin- 
guishable from  the  holohedrons  by  their  shapes,  i.e.,  they  are  geo- 
metrically identical  with  the  corresponding  holohedrons.) 


Hemihedrons 


Jtioionearons 

Gyroidal 

Pentagonal                Tetrahedral 

i 

mOn 

mOn 

—  ] 

TwOw]                   mO 

i                              i 

n 

r.i. 

±                            ± 

2 

2      J                                  2 

mO  oo 

mO  oo 

mO  oo  1 
±                                mO 

oo 

mOm 

mOm 

mOm 

mOm                ± 

2 

mO 

mO 

mO 

mO                  ± 

2 

ooQ 

ooO 

ooO                            ooO 

ooO  oo 

ooO  oo 

ooO  oo                       ooO 

00 

O 

0 

o              ±— 

%                                                      2 

Combinations  of  Hemihedrons. — Hemihedrons  combine 
with  each  other  exactly  as  do  holohedrons.  They  may  combine 
with  other  hemihedrons  of  the  same  grade  of  symmetry,  but  not 
with  those  of  different  grades  of  symmetry.  The  combining 


52  GEOMETRICAL    CRYSTALLOGRAPHY 

forms  may  all  be  hemihedrons  with  distinctive  forms  different 
from  holohedral  forms,  or  they  may  be  in  part  hemihedrons  that 
are  geometrically  identical  with  holohedrons,  provided  the  latter 
are  not  such  as  may  yield  new  hemihedrons  with  the  grade  of 
symmetry  of  the  combining  forms.  In  other  words,  the  hemihe- 
drons in  each  vertical  column  of  the  above  table  may  be  found  in 
combination,  but  not  those  in  different  columns,  except  when  the 

forms  are  alike.     Thus,  O  cannot  combine  with  —    — ,  but    oo  O 

2 

and  ooQoo  may  be  found  in  combination  with  any  hemihedron. 
Further,  mOm,  mO,  and  O  are  never  found  in  combination  with 
tetrahedral  hemihedrons,  nor  mO  oo  with  pentagonal  ones.  When 
the  +,  the  — ,  the  right,  or  the  left  forms  alone  occur  in  combi- 
nation there  is  no  difficulty  in  distinguishing  them  from  the  cor- 
responding holohedral  forms,  since  by  counting  the  planes 
possessing  the  same  symbol  it  may  be  learned  whether  they  are 
sufficiently  numerous  to  constitute  the  holohedral  form  or  only 
one-half  this  number. 

By  inspection  of  the  crystal  represented  in  Fig.  54  we  detect 
two  kinds  of  planes,  A  and  B.  The  symbol  of  the  A  planes  is 
ooa  :  a  :  ooa;  that  of  the  B  planes  is  ma  :  a  :  oca.  There  are 
six  of  the  former  present  on  the  crystal  and  12  of  the  latter.  The 
holohedral  form  ooO  oo  possesses  six  planes;  the  holohedral  form 
mO  possesses  24.  Hence  the  crystal  is  a  combination  of  oo  O  oo 

mO. 

and  ±- 
2 

When  similar  +  and  — ,  or  right  and  left  forms  are  in  com- 
bination with  one  another,  or  when  they  are  both  present  in 
combination  with  other  forms,  it  becomes  more  difficult  to  dis- 
tinguish between  them  and  the  holohedral  form  from  which  they 
are  derived,  for  by  a  combination  of  the  two  hemihedrons  all 
the  planes  of  the  original  holohedron  are  represented,  and  it  is 
only  by  a  difference  in  size  of  the  planes  of  the  two  hemihe- 
drons or  by  some  difference  in  their  appearance  that  their  true 
nature  is  recognized. 

Figure  55  represents  the  combination  of   +      and   -      .     If 


HEMIHEDRISM    OF    THE    ISOMETRIC    SYSTEM 


53 


the  planes  of  these  two  forms  were  of  equal  size,  there  would  be 
no  geometrical  difference  between  the  combination  and  the 
holohedral  octahedron.  The  fact  that  four  of  the  planes  are 
small  and  four  are  large  and  that  the  small  planes  occupy  the 
positions  of  the  planes  of  one  tetrahedron  while  the  large  ones 
occupy  the  positions  of  the  planes  of  its  congruent  form,  serves 
as  the  criterion  by  which  this  combination  is  distinguished  "from 
the  corresponding  holohedron. 


FIG.  54. — Combination  of 
ooQoo  and  +-. 


FIG.  55. — Combination  of 
+  0and_0. 


Tetartohedrism  of  the  Isometric  System  (Tetrahedral 
Pentagonal  Dodecahedral  Class). — Only  four  new  tetartohedral 
forms  are  possible  in  the  isometric  system,  and  these  are  rare. 
They  are  derived  from  the  hexoctahedron  by  the  development  of 
three  alternate  planes  in  each  alternate  octant.  They  are  known 
as  positive  and  negative,  right  and  left  tetrahedral  pentagonal 

,    ,       ,     ,  ™    .  mOn  mOn 

dodecahedrons.     Their  symbols  are  ± >—   -  and  ±  /•—  — . 


CHAPTER  VII. 


THE  HEXAGONAL  SYSTEM. 

Systems  with  One  Principal  Plane  of  Symmetry. — The 

hexagonal  and  the  tetragonal  systems  of  crystals  are  characterized 
by  possessing  holohedrons  with  one  principal  plane  of  symmetry 
and  several  secondary  planes.  The  principal  plane  is  perpen- 
dicular to  the  secondary  planes  (Figs.  56  and  57),  all  of  which 
intersect  in  a  common  line.  The  differences  in  the  geometrical 
forms  belonging  to  the  two  systems  arise  from  the  presence  of  four 
secondary  planes  of  symmetry  in  the  tetragonal  crystals  and  six 
in  the  hexagonal  crystals. 

In  each  system  the  line  of  the  intersection  of  the  secondary 
planes  of  symmetry  is  taken  as  one  of  the  crystallographic  axes, 


FIG.  56.  N  FIG.  57. 

Distribution  of  planes  of  symmetry  in  systems  with  one  principal  plane. 

and  the  intersections  of  the  principal  plane  of  symmetry  with 
alternate  secondary  planes  give  the  other  axes.  In  each  case 
the  first  axis  differs  from  the  others  which  are  all  equal.  The 
two  systems  agree  in  possessing  one  axis  which  is  not  inter- 
changeable with  the  others.  In  studying  the  crystals  the  latter 
is  always  held  in  a  horizontal  position.  The  analogies  existing 
between-  the  hexagonal  and  tetragonal  system  are  so  close  that 
a  careful  study  of  one  makes  the  study  of  the  second  very  easy. 

54 


THE  HEXAGONAL   SYSTEM  55 

Symmetry  of  the  Hexagonal  System. — The  hexagonal 
system  includes  all  crystallographic  forms  possessing  one  principal 
plane  of  symmetry  and  six  secondary  planes  and  all  the  hemi- 
hedral  and  tetartohedral  forms  that  may  be  derived  from  these. 
The  secondary  planes  intersect  each  other  in  a  common  line 
and  at  an  inclination  of  30°.  The  principal  plane  of  symmetry 
is  perpendicular  to  these  secondary  planes  (see  Fig.  56). 

Crystallographic  Axes. — The  lines  chosen  as  the  axes  of  the 
system  are  the  intersection  of  the  secondary  planes  with  each 
other  and'  the  intersection  of 

i    « 

alternate  secondary  planes  with 
the  principal  plane  of  sym- 
metry. This  selection  yields 
three  lines  inclined  to  each  _„  f 
other  at  angles  of  60°,  and  all 
perpendicular  to  a  fourth  line. 
The  latter  is  called  the  vertical 
axis  and  the  other  three  the 
lateral  axes  (Fig.  58). 

rr^,  i        /•  .M  ,•      i        FIG.   <$8. — The  system  of  axes  in  the 

The  tWO  ends  Of  the  Vertical  hexagonal  system. 

axis  are  separated  by  a  plane 

of  symmetry,  hence  these  two  ends  must  be  equivalent.  The 
lateral  axes  are  also  separated  by  planes  of  symmetry  bisecting 
the  angles  between  them,  consequently  the  lateral  axes  must  all 
be  equivalent.  But  no  plane  of  symmetry  lies  between  the 
vertical  and  the  lateral  axes.  These,  therefore,  are  not  equivalent ; 
consequently,  while  the  unities  on  the  three  lateral  axes  must  be 
equal,  the  unity  on  the  vertical  axis  has  a  different  value. 

Designation  of  the  Axes. — As  has  already  been  explained, 
in  studying  crystals  of  the  hexagonal  system  the  lateral  axes  are 
held  horizontally.  The  vertical  axis  thus  becomes  upright. 
The  former  are  designated  by  the  letter  a,  and  the  latter  by  the 
letter  c.  The  scheme  of  the  axes  is  a  :  a  :  a  :  c.  This  symbol 
indicates  that  the  unities  on  the  three  lateral  axes  are  the  same 
and  that  the  unity  on  the  vertical  axis  is  of  some  other  value. 
The  signs  given  to  the  axes  are  indicated  in  figure  58. 

A  model  representing  the  ratios  of  the  unity  lengths  of  the 


50  GEOMETRICAL   CRYSTALLOGRAPHY 

axes  would  be  constructed  of  three  straight  wires  of  equal  length 
intersecting  each  other  at  angles  of  60°,  and  all  perpendicular 
at  their  point  of  intersection  to  a  fourth  wire  of  different  length. 

The  Groundform  and  Axial  Ratio. — The  groundform  of 
the  system  is  composed  of  planes  cutting  three  of  the  axes  at 
distances  that  are  relatively  the  same  as  the  unity  distances. 
Such  planes  cut  two  of  the  lateral  axes  at 
precisely  the  same  distance  from  their  point 
of  intersection,  and  the  third  axis  -(which 
is  the  axis  c)  at  a  longer  or  shorter  distance 
from  this  point.  (See  Fig.  59.)  The  third 
lateral  axis  is  cut  at  some  distance  other 
than  unity — a  distance  which  for  the  pres- 
ent we  may  represent  as  x.  The  symbol 
of  a  plane  of  this  kind  is  a  :  a  :  xa  :  c. 
FIG.  59.— Groundform  in  When  one  of  these  planes  is  present,  sym- 

hexagonal  system. 

metry  demands  the  presence  of  n  others, 
which  together  form  a  double  hexagonal  pyramid  (Fig.  59). 

Since  the  unity  length  of  c  is  different  from  the  unity  length  of 
the  a  axes,  it  becomes  necessary,  before  an  hexagonal  crystal  can 
be  studied,  to  determine  the  ratio  between  the  two  unities  in  order 
that  a  standard  may  be  obtained  to  which  to  refer  the  intercepts  of 
other  planes  on  c.  This  unity  is  always  recorded  in  terms  of 
the  unity  on  a.  It,  therefore,  represents  a  ratio  between  the 
lengths  at  which  a  plane  of  the  groundform  cuts  one  of  the  lateral 
axes  and  the  length  at  which  it  cuts  the  vertical  axis. 

Because  it  is  a  ratio  between  standard  lengths  on  the  axes,  it  is 
known  as  the  axial  ratio,  and  is  written  in  the  form  of  a  ratio  as 
a  :  c  =  i  :  1.0999.  This  means  that  the  unity  on  c  is  1.0999 
times  the  length  of  the  unity  on  a.  The  value  of  the  axial  ratio 
depends  primarily  upon  the  groundform  chosen,  as  the  inclination 
of  the  planes  of  the  groundform  to  the  a  and  the  c  axes  determines 
the  ratio  between  the  intercepts  on  these  axes. 

Let  ABC  and  A'B'C'  (Fig.  60)  be  the  planes  of  two  ground- 
forms  cutting  the  axes  OA,  OB,  and  OC  at  A,B,C,  and  A',B',C, 
respectively.  The  ratio  between  the  lengths  on  OA  and  OB  and 
on  OC  will  be  determined  by  the  inclination  of  the  planes  to  the 


THE  HEXAGONAL    SYSTEM 


57 


axes — the  larger  the  angle  made  between  the  plane  and  the 
lateral  axes,  the  larger  will  be  the  ratio  between  the  intercepts  on 
these  axes  and  that  on  c;  or  the  larger  will  be  the  axial  ratio  or, 
in  other  words,  the  larger  will  be  the  unity  on  c. 

Determination  of  the 
Axial  Ratio. — Every  sub- 
stance that  crystallizes  in  the 
hexagonal  system  possesses  a 
different  groundform.  This 
groundform  is  always  a  double 
hexagonal  pyramid,  but  the 
inclination  of  the  faces  differs 
for  every  different  substance. 
Hence,  for  every  hexagonal 
mineral  the  axial  ratio  must 
be  determined  before  the  sym- 
bols of  the  planes  occurring 

on  its  crystals  can  be  calculated.  Practically  a  form  whose 
planes  intercept  the  vertical  axis  and  two  of  the  lateral  axes  is 
assumed  as  the  groundform.  The  relative  distances  at  which 
one  of  its  planes  cuts  the  a  and  the  c  axes  are  calculated,  after 

x 


FIG.  60. 


measurement  of  the  proper  interfacial  angles,  and  this  ratio  be- 
comes the  axial  ratio  which  is  accepted  by  all  crystallographers 
as  representing  the  arbitrary  value  which  shall  be  regarded  as 
the  ratio  between  the  unities  on  these  axes. 


GEOMETRICAL   CRYSTALLOGRAPHY 


In  figure  61  let  A  represent  the  groundform  of  a  crystal 
the  axial  ratio  of  which  is  to  be  determined,  and  DEF,  the  inter- 
facial  angle  between  two  faces  measured  at  the  termination  of 
the  line  OE,  drawn  from  the  point  of  intersection  of  the  axes 
perpendicular  to  the  lateral  edge.  The  distance  OX  =  unity  on 
c,  and  OK  unity  on  a.  Let  B  be  a  section  through  the  lines 
OE  and  OX,  and  C,  a  section  through  the  lateral  axes.  Repre- 
sent one-half  the  measured  angle  by  /?. 

In  the  triangle  OEX,OX=OE  tan  p  (i) 

In  the  triangle  OEK,OE  =  OK  sin  OKE 

Since  OK  =  i  and  OKE  =  60°,  and  sin.  60°  =  .  866 

OE=.866 
Substituting  in  (i)  we  have  OX  or  c—  .  866  tan  /? 

That  is,  the  natural  tangent  of  one-half  the  measured  lateral 
interfacial  angle  on  the  ground-form  pyramid  multiplied  by  .  866 
is  the  axial  ratio.  In  the  mineral  quartz  (SiO2) 
the  angle  between  the  planes  p  and  r  is  141° 
47'  (Fig.  62).  From  this  value  the  angle  between 
p  and  z  (below)  is  easily  calculated  as  103°  34'. 
One-half  of  this,  or  /?,  =51°  47',  and  its  natural 
tangent  is  1.27.  Substituting  in  the  equation 
above  we  have  i.  27 X  .866=1.0998,  which  is 
the  axial  ratio  for  all  crystals  of  quartz. 

There    are    thus    as    many    different    axial 
ratios  as  there  are    substances  crystallizing  in 
the  system.     Consequently  the  axial  ratio  is  a 
most  important  distinguishing  characteristic  of 
hexagonal  substances.     For  nine  important  hexagonal  minerals 
it  is  as  follows: 

Quartz  i  :  1.0999    Dolomite  i  :     .8322    Nepheline     i  :  .8389 
Calcite   i  :     .8543    Cinnabar  i  :  1.1453    Beryl  i  :  .4988 

Apatite  i  :     .7346    Hematite  i  :  1.3650    Tourmaline  i  :  .4481 

The  Intercept  on  the  Third  Lateral  Axis. — The  ground- 
form  in  the  hexagonal  system  is  a  double  pyramid  each  of  whose 
planes  cuts  two  of  the  lateral  axes  at  the  same  distance,  assumed 
as  unity,  and  the  vertical  axis  at  some  different  distance  which 


FIG.  62. — Crystal  of 
quartz. 


THE  HEXAGONAL    SYSTEM  59 

is^also  taken  as  unity.  The  symbol  of  its  plane  as  given  above 
(page  56)  is  a  :  a  :  xa  :  c.  As  usually  written  the  symbol  is 
a  :  a  :  c,  or  P. 

The  intercept  on  the  third  axis  is  generally  omitted  from  the 
symbol  because  its  value  is  determined  by  the  intercepts  on  the 

n 

other  lateral  axes,  in  such  a  way  that  x  is  always  -  — ,  where  n  = 

n  —  i 

the   intercept  on  the  first  a  axis.     In  the  symbol  a  :  a  :  xa  :  c, 

n=  i  and  x=-=  <*>.     The  plane  is  parallel  to  the  third  a  and  its 
o 

complete  symbol  is  a  :  a  :  ooa  :  c.  The  oo  in  the  symbol  is 
omitted  because  if  the  intercepts  on  two  of  the  a  axes  are  known 
the  intercept  on  the  third  axis  is  also  known. 

In  using  the  indices  to  designate  forms  in  this  system  the 
alternate  ends  of  the  axes  are  considered  positive  and  the  interven- 
ing ends  negative  as  indi- 
cated in  figure  58.  The 
intercepts  are  written  in  the 
order  +al  +a2,  —  a3,  c,  and 
the  symbol  becomes  hikl. 
In  all  symbols  of  hexagonal 
forms  the  sum  of  the  indices 
on  the  three  lateral  axes 

1  7       ,     •     ,     L  FlG-    63- 

always  =  zero;  i.e.,. «'•+*:+£ 

=  o.     Consequently,  if  two  of  the  indices  are  known,  the  third 

can  easily  be  deduced. 

Proof  that  the  Intercept  on  the  Third  Lateral  Axis  is 

n 
— ,  when  the  Intercepts  on  the  Other  Two  Lateral  Axes 

n  —  i 

are  n  and  Unity.— In  figure  63  let  AO,  BO,  and  OC  represent 
the  lateral  axes  intersecting  at  O  in  the  center  of  a  crystal,  and 
let  GH  be  the  intersection  of  a  pyramidal  face  with  the  plane  of 
these  axes.  This  plane  intercepts  the  three  axes  at  OG,  OB,  and 
OH,  respectively,  and  OG,  OB,  and  OH  are  its  parameters  on 
these  axes.  If  OG=n,  and  OB  =  unity  =i,  then  OH,  the 

n 

parameter  on  the  third  axis,  is  -  — . 

n — i 


60  GEOMETRICAL    CRYSTALLOGRAPHY 

Inscribe  a  circle  with  O  as  the  center  and  a  radius  equal  to  OB. 
Then  will  OA,  OB,  OC,  etc.,  be  the  unity  distances  on  these  axes. 

Draw  the  line  B A  from  the  unity  distance  on  B  O  to  the  unity 
distance  on  AO.  In  the  triangle  AOB,  the  sides  OA  and  OB  are 
equal,  .  *  .  the  angles  OAB  and  OBA  are  equal  and  each  is  60°. 
The  angle  GAB  is  therefore  120°,  as  is  also  the  angle  AOH;  .  •  . 
AB  is  parallel  to  OH.  Moreover,  the  triangle  is  equilateral  and 
AB  =  OB  =  i.  The  triangles  GAB  and  GOH  are  similar  .  •  . 
GA  :  GO  :  :  AB  :  OH,  or  GA  :  n  :  :  i  :  OH.  But  GA=GO- 
AO=n  —  i,  .  *  .  the  equation  becomes n  —  i  :n  :  :  i  :  OH.  Mul- 

n 

tiplying  extremes  and  means  gives  OH  X  n  —  i  =n,  or  OH  = . 

n  —i 

OH  is  the  intercept  on  the  axis  OC,  thus  the  parameter  on  the 

third  axis  is  ,  when  the  parameters  on  the  other  two  axes  are 

n  —  i 

n  and  i.  . 

Symbols  of  Forms. — In  this  system  and  in  all  of  the  succeed- 
ing systems  the  groundform  is  represented  by  P.  Unity  inter- 
cepts are  not  represented.  Other  intercepts  on  the  c  axis  are 
written  in  front  of  P,  and  one  of  the  lateral  intercepts,  provided 
these  are  not  unity,  is  written  after  the  P.  In  practice  the  smaller 
of  the  two  lateral  intercepts,  when  they  are  not  equal,  is  indicated, 

the  larger  one  being  understood.     Thus  -a  :  a  :  $a  :  $c  is  written 

3  P  -,  it  being  understood  that  when  one  lateral  intercept  is  - 
4  4' 

the  other  is  5. 

HOLOHEDRAL  DIVISION. 

(Dihexagonal  Bipyramidal  Class.) 

Symmetry  of  the  Holohedral  Forms. — The  holohedral 
forms  of  the  hexagonal  system,  like  the  holohedral  forms  of  the 
isometric  system,  possess  the  symmetry  which  is  described  as 
characterizing  the  system.  In  addition  .to  the  planes  of  symmetry 
described  (Fig.  56)  the  holohedrons  of  this  system  possess  a 
sixfold  axis  of  symmetry  perpendicular  to  the  principal  plane 


THE  HEXAGONAL   SYSTEM 


6l 


of  symmetry  and  six  axes  of  binary  symmetry  at  right  angles  to 
the  sixfold  axis,  and  a  center  of  symmetry  (Fig.  64).  Figure  65 
exhibits  the  symmetry  relations  projected  on  the  plane  passing 
through  the  lateral  axes,  i.e.,  upon  the  principal  plane  of 
symmetry. 

The  Most  General  Form. — The  most  general  form  of  the 
system  is  composed  of  planes  which  cut  the  c  axis  at  some  dis- 
tance other  than  unity  and  the  three  lateral  axes  at  different 


FIG.  64. 


FIG.  65 — Diagram  illustrating  the 
distribution  of  the  elements  of  sym- 
metry in  hexagonal  holohedrons. 


distances.     Since  one  of  the  latter  may  be  considered  as  unity, 
the  Naumann  symbol  of  a  plane  of  the  most  general  form  is 

na  :  a  :          a  :  me,  where  n  is  greater  or  less  than  -  — ,  provided 
n—  i  n  —  i 


neither   is  unity.     When    either  n   or 


n 


n  —  i 


is  unity,   or  when 


n 


n= ,  the  plane  will  cut  two  of  the  lateral  axes  at  the  same 


n  —  i 


distance,  and  hence  cannot  belong  to  the  most  general  form  in 
the  system. 

\    n 

The  Dihexagonal  Bipyramid. — Whenw  ^-    — ,  and  neither  is 


n  —  i 


unity,  the  planes  cut  one  of  the  lateral  axes  at  unity,  another  at 


n  and  the  third  at 


The  presence  of  one  of  these  planes 


62  GEOMETRICAL    CRYSTALLOGRAPHY 

necessitates  the  presence  of  1 1  others  on  one  side  of  the  principal 
plane  of  symmetry,  and  twelve  on  its  opposite  side.  In  all  there 
are  24  planes  on  the  form,  one  in  each  of  the  24  compartments 
into  which  the  symmetry  planes  of  the  system  divide  space. 

The    form  thus  produced    is    a  double    pyramid   (Fig.   66) 
bounded  by  24  similar  scalene  triangles.     It  is  known  as  the  di- 
hexagonal bipyramid.     The  symbol  of  one  of  its 

planes  is  na  :  a  :  a  =  me,  or  if  the  c  axis  is 

n  —  i 

cut  at  the  distance  corresponding  to  the  axial 

ratio,  the  symbol  is  na  :  a  :          a  :  c,  and  the 

n  —  i 

symbol  of  the  form  is  mPn,  or  Pn.     Its  indices 

FIG.  66.— Dihex-    are  ^-     As  has  alread7  been   stated    (page 
agonal    bipyramid,    ^Q)  ^  it  makes  no  difference  whether  the  shorter 
mPn  or  hikl.        Qr  the  ionger  of  tjie  two  parameters  that  refer 
to  the  a  axis  is  written  after  the  P  in  the  symbols  of  the  forms 
in  this  system,  as  the  value  of  each  of  these  parameters  depends 

upon  the  value  of  the  other.     The  symbol  2  P  --  and  2  P  3  repre- 

n 

sent  exactly  the  same  form,  for  n  necessitates      — ,  and  wee  -versa. 

n  —  i 

If  n  =  3,  then   "    =3;  if  n  =  -,  then    '      =     r  =  3-     It  is  simply 
n-i     2  2  n-i      1/2 

custom  that  requires  the  use  in  the  symbol  of  the  smaller  of  the 
two  parameters  that  are  not  unity. 

The  Dihexagonal  Series. — A  series  of  forms  embraces  those 
that  may  be  derived  from  each  other  by  a  change  in  the  value  of 
a  single  parameter.  The  dihexagonal  series  is  the  series  of  forms 
that  may  be  derived  from  the  dihexagonal  bipyramid  by  changing 
the  parameter  on  the  c  axis. 

The  symbol  of  a  plane  of  the  most  general  form  is  na  :  a  : 

—a  :  me.     By  changing  the  parameter  on  c  we  may  obtain  the 
n  —  i 

following  symbols: 

n 

(i)  na  :  a  :          a  :  me. 
n  —  i 


THE  HEXAGONAL   SYSTEM  63 

U 

(2)  na  :  a  :          a  :  c. 
n  —  i 

n          i 

m  na  :  a  :      — a  :  —c. 
n-i       m 

n 

(4)  na  :  a  :          a  :  <*>c. 

(5)  na  :  a  :          a  =  oc. 

The  planes  represented  by  these  symbols  all  bear  the  same 
relation  to  the  a  axes.  They  are,  however,  differently  inclined 
to  the  c  axis,  and  so  produce  forms  that  differ  from  the  dihex- 
agonal  bipyramid  according  to  the  distance  at  which  their  planes 
cut  the  vertical  axis. 

The  planes  represented  by  the  symbols  (i),  (2)  and  (3)  are  di- 
hexagonal  bipyramids.  When  either  (2)  or  (3)  occurs  alone  it 
cannot  be  distinguished  from  the  most  general  form  (i)  unless 
the  value  of  the  unity  on  the  c  axis  is  known.  When,  however, 
this  unity  is  known,  a  calculation  of  the  values  of  the  intercepts 
on  c  will  discriminate  between  the  three  forms. 

For  example,  one  may  have  three  dihexagonal  bipyramids  of 
the  same  substance.  Measurement  of  their  interfacial  angles 
may  show  that  a  plane  on  one  cuts  the  c  axis  at  2 . 1998  times  the 
shortest  distance  at  which  it  cuts  the  a  axes,  which  distance  is 
taken  as  the  unity  on  these  axes;  that  a  plane  on  the  second  crystal 
intercepts  the  c  axis  at  1.0999  times  the  unity  on  a;  and  that  a 
plane  on  the  third  crystal  intercepts  c  at  .  54995  times  this  unity 
on  a.  Evidently  the  three  forms  to  which  these  planes  belong  are 
different  and  should  be  represented  by  different  symbols.  But 
there  is  nothing  on  the  crystals  themselves  to  guide  us  in  the 
selection  of  the  symbols.  If,  however,  we  know  that  the  crys- 
tallized substance  is  quartz,  the  solution  of  the  difficulty  is  easy. 
The  axial  ratio  of  quartz  is  i  :  1.0999;  i-e-> the  accepted  relation 
between  the  unities  on  a  and  c  is  as  i  :  1.0999.  The  first  of  the 
three  forms  discussed  is  composed  of  planes  that  cut  the  c  axis  at 
twice  this  ratio,  hence  its  symbol  is  2  Pn.  The  symbol  of  the  form 
on  the  second  crystal  is  Pn,  and  that  on  the  third  crystal,  1/2  Pn. 


64 


GEOMETRICAL   CRYSTALLOGRAPHY 


The  Dihexagonal  Prism, — The  fourth  symbol,  na  :  a 


—  a 
n—i 

:  oo  c  represents  a  plane  that  differs  from  a  plane  of  mPn  in  that 
it  is  parallel  to  the  c  axis.  The  resulting  form  is  a  1 2-sided  prism 
(Fig.  67),  the  cross  section  of  which  is  the  same  as  the  cross  section 
of  the  dihexagonal  bipyramid  along  the  principal  plane  of  sym- 
metry. Its  symbol  is  &Pn  and  its  indices  are  hiko. 

Basal  Pinacoid. — The  fifth  symbol,  na  :  a  :  a:  oc,  stands 

n  —  i 

for  a  plane  that  cuts  the  c  axis  at  its  point  of  intersection  with  the 
other  axes,  and  the  a  axes  all  along  their  lengths.  In  other  words, 

it  is  a  plane  parallel  to  the  principal  plane  of 

symmetry.     It  is  known  as  the  basal  pinacoid. 

Its  symbol  is  oP,  and  indices,  oooi.. 

The  symbol  of  this  pinacoid  requires  that 

the  plane  shall  cut  one  lateral  axis  at  unity 

and  the  other  two  at  n  and  -  — ,  respectively, 

n  —  i 

and  at  the  same  time  that  it  shall  pass  through 
the  intersection  of   the   axes.     A   plane  that 
FIG.   67.— Dihex-  cuts  either  of  the  lateral  axes  at  any  distance, 

agonal  prism,  ooPw  or  ....  .       .     .      . 

provided  it  passes  through  their  intersection 


with  the  c  axis,  must  cut  them  at  all  dis- 
tances, hence  it  is  not  necessary  to  specify  in  the  symbol  any 
definite  distance  at  which  it  cuts  these  a  axes,  since  all  distances 
are  understood. 

The  basal  pinacoid  is  a  pair  of  planes  perpendicular  to  the  c 
axis.  Two  of  them  comprise  the  form,  since  the  existence  of 
one  on  either  side  of  the  plane  of  symmetry  necessitates  the 
existence  of  one  more  on  its  opposite  side. 

In  nature  trie-planes  do  not  actually  pass  through  the  center  of 
the  crystals,  since  this  is  a  physical  impossiblity.  They  are  found 
at  the  terminations  of  the  c  axis,  being  parallel  to  the  imaginary 
plane  passing  through  the  center  of  the  crystal.  (See  Fig.  73.) 

Three  Series  of  Holohedrons  in  the  Hexagonal  System.— 
The  three  forms  already  noted,  viz.,  the  dihexagonal  bipyramid, 
and  prism,  and  the  basal  plane,  constitute  the  dihexagonal 


THE  HEXAGONAL   SYSTEM 


series.     Besides  this  series  there  are  two  others  possible  in  the 
system. 

The  dihexagonal  series  results  when  all  of  the  lateral  axes  are 
cut  at  different  distances.  When  two  of  the  lateral  axes  are  cut 
at  the  same  distance,  two  other  series  arise,  according  to  whether 

n  n 

in  the  general  symbol  na  :  a  :      —a  :  me,  n  =  —  — ,  or  either  n,  or 

n— i  n— i 

n 

=  unity. 
n  —  i 

When  n  =  i ,         •  =  oo ;  and,  conversely,  when =  i ,  n  =  <x> . 

n  —  i  n  —  i 

n 

When  n  = ,n=2:  for  clearing  the  equation  of  fractions  we 

n  —i 

have  n(n  —  i)=n.     Dividing  by  n  this  is  reduced  ton  —  i=i,  when 
n=2. 

These  two  changes  in  the  general  symbol  are  the  only  two  that 
can  possibly  give  rise  to  new  forms  from  which  new  series  may 
be  derived  by  changes  in  the  parameters  on  the  c  axis. 

Pyramids  and  Prisms  of  the  First  Order. — A  plane  with  the 
symbol  a  :a  :  o>a  :  me,  cuts  two  of  the  lateral  axes  at  unity  and  the 
c  axis  at  some  other  distance  than  the 
unity  on  this  axis.  The  third  lateral  axis 
is  cut  at  oo.  When  one  plane  with  this 
symbol  is  present,  symmetry  requires  the 
presence  of  eleven  other  similar  planes 
(Fig.  68),  resulting  in  a  twelve-sided 
bipyramid  composed  of  six  isosceles  tri- 
angular faces  above  the  principal  plane 
of  symmetry,  and  the  same  number  of 
similar  faces  below  it.  It  is  known  as  ~TO 

FIG.  68. — Hexagonal  ra- 
the hexagonal  bipyramid.  Its  symbol  is  pyramid  of  the  first  order, 
mP.  The  indices  are  ho  hi.  »P  or  h<*i. 

From  an  inspection  of  the  cross  section  of  this  pyramid  it  will 
be  noted  that  the  lateral  axes  terminate  in  the  solid  angles  between 
four  contiguous  pyramidal  faces.  Forms  that  bear  this  relation 
to  the  axes  are  known  as  forms  of  the  first  order. 


66 


GEOMETRICAL   CRYSTALLOGRAPHY 


From  this  pyramid  a  series  of  pyramids  and  a  prism  are  derived 
by  changing  the  intercept  on  c,  as  follows : 


(I) 

a  :  a  :   a 

3  a  :  mc  —  mP 

(2) 

a  :  a  :  a 

>a  :     c  =  P 

i        i 

(3) 

a  :  a  :   a 

>a;-i-c=-  P 

m       w 

(4) 

a  :  a  :  a 

•  a  :  ooc=  a>P 

(5) 

a  :  a  '.  & 

>#  :   oc  =  oP 

The  symbols  1,2,  and  3  represent  pyramids  of*the  first  order 
that  differ  from  each  other  in  the  relative  lengths  of  their  vertical 


FIG.  6g. — Hexagonal  prism  of 
the  first  order,  ooP  or  lolo. 


FIG.  70. — Hexagonal  bipyramid  of 
the  second  order,  mP2,  or  hbzhl. 


dimensions.     Symbol  (2)  represents  the  groundform  of  the  sys- 
tem.    The  pyramid  mP  is  more  acute  than  this,  and  the  pyramid 

— P  more  blunt. 
m 

Symbol  (4)  represents  a  prism  of  the  first  order  (Fig.  69). 
This  is  composed  of  six  faces  with  the  axes  terminating  in  their 
interfacial  edges.  Its  symbol  is  c»P  or  1010. 

The  plane  a  :  a  :  oo#  :  oc  does  not  differ  in  character  from 

H 

the  plane  na  :  a  : a:  oc,  both,  therefore,  are  represented   by 

the  same  symbol;  i.e.,  oP. 

Pyramids  and  Prisms  of  the  Second  Order. — The  symbol 
2a  :  a  :  2a  :  me,  or  h  h  2h  I,  designates  a  plane  that  cuts  two  of 
the  lateral  axes  at  twice  unity,  a  third  at  unity,  and  the  c  axis  at  m. 


THE   HEXAGONAL    SYSTEM 


67 


This  plane  belongs  to  a  bipyramid  of  12  faces  that  is  identical 
in  appearance  (see  Fig.  70)  with  the  hexagonal  bipyramid  of  the 
first  order.  The  difference  between  the  two  pyramids  is  simply 
in  their  relation  to  the  axis.  Whereas  in  the  form  of  the  first  order 
the  axes  terminate  in  the  solid  angles  made  by  the  conjunction  of 
four  faces,  in  the  pyramid  of  the  second  order  they  terminate  in 
the  centers  of  the  lateral  edges. 

From  the  most  general  symbol  of  the  series  four  other  symbols 
are  derived  by  changing  the  parameter  on  c  : 


(I) 

2a 

:  a  : 

2a 

:  me 

=  ml 

(2) 

2a 

:  a  : 

2a 

:  c 

=  P2 

i 

T 

(3) 

2a 

:  a  : 

2a 

=    _  I 

'  m 

m 

(4) 

2a 

:  a  : 

2a 

:  ooc 

',  ~~~    QO  J 

(5) 

2a 

:  a  : 

2a 

:  oc 

=  oP 

Symbols    (i),    (2),    (3),    represent   three    bipyramids   of   the 
second  order,  in  which  the  intercepts  on  the  vertical  axis  are 


u~ 


FIG.  71. — Hexagonal 
prism  of  the  second 
order,  coP2  or  1120. 


FIG.  72. — Diagram  illustrating  the  rela- 
tions of  holohedral  bipyramids  and  prisms 
to  the  axes  in  the  hexagonal  system. 


different.  Symbol  (4)  represents  an  hexagonal  prism  of  the 
second  order  in  which  the  axes  terminate  in  the  centers  of  the 
faces  (see  Fig.  71),  and  symbol  (5)  represents  the  basal  pinacoid. 
The  relations  of  the  pyramids  of  the  three  series  to  the  lateral 
axes  are  seen  in  the  diagram  (Fig.  72),  which  represents  cross 
sections  through  the  pyramids  of  the  three  orders  in  the  plane  of 
the  lateral  axes.  The  inner  hexagon  corresponds  to  forms  of  the 
first  order,  the  outer  one  to  those  of  the  second  order,  and  the 


68 


GEOMETRICAL   CRYSTALLOGRAPHY 


dodecagon  between  the  two  hexagons  to  the  dihexagonal  forms. 
The  three  solid  lines  intersecting  at  the  center  of  the  figure  are 
the  axes. 

In  practice  the  distinction  between  forms  of  the  first  and 
second  orders  can  be  made  only  after  the  position  of  the  axes 
has  been  determined.  The  selection  of  the  axes  is,  moreover, 
merely  a  matter  of  convenience  in  many  instances,  those  lines 
being  chosen  as  the  axes  which  will  yield  the  simplest  symbols 
for  the  forms  occurring  on  the  crystal  under  investigation. 
When  the  axes  are  once  chosen,  however,  they  remain  fixed  for  all 
crystals  of  the  substance  and  all  planes  must  be  referred  to  them. 


FIG.  73. — Hexagonal 
prism  (coP)  terminated 
by  basal  plane  (oP). 


FIG.  74. — Combina- 
tion of  hexagonal  prism 
and  bipyramid  of  same 
order. 


FIG.  75. — Combina- 
tion of  hexagonal  prism 
and  bipyramid  of  dif- 
ferent orders. 


Closed  and  Open  Forms. — While  each  one  of  the  several 
forms  belonging  to  the  isometric  system  will  alone  enclose  space, 
this  is  not  true  of  all  the  forms  of  the  hexagonal  system.  The 
bipyramids  may  occur  alone  on  a  crystal,  since  each  one  com- 
pletely encloses  space.  The  prisms  and  the  basal  plane  differ 
from  the  bipyramids  in  that  neither  can  completely  enclose 
space  and  therefore  neither  can  exist  alone.  They  occur  either 
in  combination  with  each  other  (Fig.  73)  or  in  combination  with 
some  other  form. 

Forms  that  enclose  space  completely  are  often  spoken  of  as 
dosed  forms,  while  those  which  do  not  completely  enclose  space 
are  known  as  open  forms.  Crystals  bounded  by  open  forms 
cannot  be  represented  by  less  than  two  symbols. 

Combinations. — Combinations  of  holohedral  forms  in  the 
hexagonal  system  are  not  as  common  as  those  of  hemihedrons. 


THE  HEXAGONAL   SYSTEM 


69 


Where  they  occur  they  are  easily  understood,  provided  it  is 
remembered  that  the  axes  determined  for  one  form  are  the  axes 
to  which  all  the  forms  in  the  combination  must  be  referred. 

Figure  74  represents  a  combination  of  a  prism  and  a  pyramid 
belonging  to  the  same  order;  figure  75,  a  prism  and  a  pyramid 


FIG.  77. — Crystal  of  beryl  con- 
taining ooP  (a),  P(p),  2P2(r), 
2P(w),  3Pf  (v),  and  oP(c). 


FIG.  76.— Combination  of 
2  hexagonal  pyramids  of  the 
same  order. 


of  different  orders;  and  figure  76,  two  pyramids  of  the  same 
order.  Figure  77  is  more  complicated.  It  represents  a  beryl 
(Be3Al2(SiO3)6)  crystal  on  which  are  the  forms  ooP  (a), 


HEMIHEDRAL  DIVISION. 

Hemihedrism  in  the  Hexagonal  System.  —  Although  the 
hemihedral  divisions  of  this  system  are  as  important  as  the 
holohedral  division,  we  must  limit  our  discussion  to  a  few  of  the 
simplest  of  the  hemihedral  forms.  These  are  extremely  impor- 
tant, since  some  of  the  commonest  minerals,  like  calcite  (CaCO3), 
dolomite  (MgCaCO3),  and  apatite  (Ca3Cl(PO4)3),  possess  them. 

Possible  Kinds  of  Hemihedrism.  —  It  will  be  remembered 
that  the  lateral  axes  are  not  equivalent  to  the  vertical  axis  in  this 
system.  Hence  the  law  of  hemihedrism  does  not  require  that 
in  the  hemihedrons  the  ends  of  the  vertical  axis  shall  be  ter- 
minated in  the  same  way  as  the  ends  of  the  lateral  axes.  It 
demands  simply  that  the  new  hemihedral  forms  shall  affect  the 
two  ends  of  the  vertical  axis  similarly  and  the  several  ends  of  the 
lateral  axes. 


70  GEOMETRICAL   CRYSTALLOGRAPHY 

The  number  of  possible  ways  by  which  one-half  the  planes  of 
the  general  form  in  this  system  may  be  combined  is  even  greater 
than  in  the  isometric  system.  In  only  four  cases,  however,  does 
the  result  comply  with  the  conditions  of  hemihedrism.  These 
are  indicated  in  the  accompanying  figures,  in  which  all  the  white 
planes  are  represented  as  surviving  or  all  the  shaded  planes. 

The  four  classes  of  hemihedrism  derived  by  these  four 
methods  possess  different  grades  of  symmetry.  The  names 
applied  to  them  are: 

The  Rhombohedral  class  (Fig.  78)  in  which  the  hemihedrons 
possess  all  the  faces  of  the  holohedrons  that  lie  in  alternate 
dodecants. 


FIG.  78.  FIG.  79.  FIG.  80.  FIG.  81. 

Figures  illustrating  the  four  possible  methods  of  derivation  of  the  hemihedrons 
in  the  hexagonal  system.  Fig.  78,  rhombohedral;  79,  pyramidal;  80,  trapezohe- 
dral;  81,  trigonal. 

The  Pyramidal  class  (Fig.  79)  in  which  the  alternate  pairs  of 
planes  meeting  in  the  lateral  edges  survive. 

The  Trapezohedral  class  (Fig.  80)  in  which  the  planes  that 
survive  occupy  the  alternate  compartments  of.  the  24  into  which 
space  is  divided  by  the  planes  of  symmetry. 

The  Trigonal  class  (Fig.  81)  in  which  all  the  planes  in  alternate 
sextants  survive. 

Only  rhombohedral  and  pyramidal  hemihedrons  have  been 
found  on  crystals  in  complete  forms  that  are  geometrically 
distinct  from  holohedrons,  but  hemimorphic  forms  of  the  trigonal 
class  are  also  known. 

Rhombohedral  Hemihedrons  (Trigonal  Scalenohedral 
Class). — The  rhombohedral  hemihedrons  may  be  regarded  as 
derived  from  holohedrons  by  the  suppression  of  all  the  planes 


THE   HEXAGONAL    SYSTEM 


71 


FIG.  82. —  Diagram 
illustrating  the  distribu- 
tion of  the  elements  of 


that  lie  in  alternate  dodecants.  The  new  forms  thus  derived 
lose  the  principal  plane  of  symmetry,  the  three  secondary  planes 
passing  through  the  lateral  axes,  and  the  three  axes  of  binary 
symmetry  lying  between  these  crystallographic  axes.  The 
remaining  elements  of  symmetry,  viz.,  the  three  secondary  planes 
lying  between  the  lateral  axes,  the  three  axes  of  binary  symmetry 
coinciding  with  these  crystallographic  axes, 
and  the  center  of  symmetry  survive.  The 
axis  of  sixfold  symmetry  becomes  an  axis 
of  threefold  symmetry.  This  coincides 
with  the  vertical  crystal  axis.  (Fig.  82.) 
In  Fig.  82,  which  is  a  diagrammatic  repre- 
sentation of  these  symmetry  relations,  the 
heavy  broken  line  indicates  the  disappear- 
ance of  the  principal  plane  of  symmetry. 
The  lighter  broken  lines  indicate  the  posi- 
tions of  the  secondary  planes  that  have 
disappeared,  and  the  light  unbroken  lines  the  position  of  those 
retained. 

Only  two  new  geometrical  hemihedrons  are  possible  in  this  class. 

The  Scalenohedrons. — The  extension  of  the  planes  of  the 
dihexagonal  pyramid  occupying  alternate  dodecants  yields  a 
form  bounded  by  twelve  similar  scalene  triangles.  (Figs.  83 
and  84.)  These  intersect  in  two  kinds  of  interfacial  edges 
extending  from  the  ends  of  the  vertical  axis  to  the  plane  of  the 
lateral  axes,  and  a  third  kind  connecting  the  ends  of  the  lateral 
axes  in  a  zig-zag  line.  The  lateral  axes  terminate  in  the  centers 
of  these  edges.  The  forms  are  known  as  hexagonal  Scaleno- 
hedrons. Their  symbols  are  ±  ——.  Two  congruent  scalenohe- 

2 

drons  may  be  derived  from  every  dihexagonal  bipyramid.  They 
are  distinguished  as  the  positive  (Fig.  83)  and  the  negative  (Fig. 
84)  forms.  Their  indices  are  *(hikl)  and  *(ihkl). 

The  Rhombohedrons. — By  suppressing  every  alternate  plane 
composing  the  pyramid  of  the  first  order  and  extending  the 
remaining  planes  two  new  congruent  forms  are  produced,  each 
bounded  by  six  similar  rhombs.  These  possess  two  kinds  of  solid 


72  GEOMETRICAL   CRYSTALLOGRAPHY 

angles,  of  which  two  are  polar  and  four  lateral.  The  polar 
angles  are  larger  or  smaller  than  the  other  four  depending  upon 
the  value  of  the  intercepts  on  c.  When  the  axial  ratio  is  i  :  \/i-5 


FIG.  83.—?  o  s  i  -  FIG.  84.— Nega- 
tive hexagonal  sea-  t  i  v  e  hexagonal 
lenohedron,  +  ™E»  scalenohedron, 
or  *(«*/).  -*f"  or  *(«*/). 


or  i  :  1.22474,  the  polar  and  lateral  angles  on  the  unity  rhom- 

/p\ 
bohedron  (  - )  are  equal.     The  new  forms  are  called  the   posi- 

\2  / 

tive    (Fig.    85)  and    the    negative    (Fig.    86)    rhombohedron— 


FIG.    85.  —  Positive  FIG.  86.  —  Negative 

rhombohedron,   +  ™   '  rhombohedron.  —  m^' 

2  a 

or  +  R,  or  K(hohl).  or  —  R,  or  K(ohtil"). 

the  latter  being  the  one  that  turns  an  upper  edge  toward  the 

observer.     Their   symbols  are  +         and  ,  and  their  in- 

2  2 


THE  HEXAGONAL   SYSTEM 


73 


dices  x(hohl)   and  i<(ohhi).     In  them  the  axes  terminate  in  the 
centers  of  the  six  lateral  edges. 

Short  Symbols  for  the  Scalenohedron  and  the  Rhombo- 
hedron. — The  two  rhombohedral  hemihedral  forms  are  so 
frequently  met  with  in  nature  that  many 
crystallographers  use  simpler  symbols  than 
those  given  above  to  represent  them. 

The  rhombohedron  derived  from  P  is 
represented  by  R.  That  derived  from  mP 
is  wR,  etc.  The  scalenohedron  is  represented 
by  R/>,  mRp,  etc.,  in  which  R  and  wR  signify 
the  rhombohedron  with  the  same  lateral  edges 
as  the  scalenohedron  (rhombohedron  of  the 
middle  edges,  Fig.  87)  and  p  the  distance  at 
which  the  scalenohedral  faces  intersect  the 
vertical  axis  in  terms  of  the  corresponding 
intersection  of  the  planes  of  the  rhombohe- 
dron of  the  middle  edges.  The  m  and  p  in 
this  symbol,  therefore,  do  not  correspond  to 

m  and  n  in  the  symbol  -  — .     The  symbols  may  be  transformed 


FIG.  87. — Hexag- 
onal scalenohedron, 
enclosing  rhombohe- 
dron of  the  middle 
edges. 


from  one  system  into  the  other  with  the  aid  of  the  following 
equations:* 

mPn     m(2  —  n)       '  n  i  P  1/2 

2  n  2  —n' 


mpP 


2p 
P  + 


Combinations. — The  combinations  of  the  pyramids  and 
prisms  of  the  first  and  second  orders  with  the  rhombohedrons  are 
at  first  confusing,  but  a  little  consideration  of  the  position  of  the 
axes  will  nearly  always  serve  to  make  them  clear. 

Figure  88  represents  a  combination  of  —  R  with  the  prism  of 

*  For  proof  of  the  correctness  of  these  equations  see  Groth:  Physikalische 
Krystallographie,  1885,  p  342  and  p.  348. 


74 


GEOMETRICAL   CRYSTALLOGRAPHY 


the  first  order;  figure  89,  a  combination  of  +R  with  the  prism  of  the 
second  order;  and  figure  90,  that  of  a  +R  with  a  —  R.     If  the 


FIG.  88.  —  Combi-           FIG.  89.  —  Com-  FIG.  90.  —  Combination  of 

nation    of     negative        bination  of  posi-  positive  and  negative  rhom- 

rhombohedron(  —  R)         tive    rhombohe-  bohedrons. 
with    prism    of    the        dron   (  +  R)  with 
first  order  (  ocP).               prism    of   the 
second   order 


positions  of  the  axes  are  fixed  with  reference  to  the  rhombo- 
hedrons  in  these  crystals,  the  nature  of  the  prisms  is  easily 
recognized.  The  crystals  represented  in  figures  91  and  92  are 


FIG.  92. — Crystal  of  cal- 
ate  containing  R(/>),|R(s), 
4R(m),R3(r)R5(;y),RH?'), 
and  R30- 


FIG.  91. — Hematite  (Fe2O3) 
crystal  containing  R(R),  — 
R(r)  and-iR(n). 


more  complicated.     The  first  contains  planes  of  the  forms 

— R(r),  and   -  --R  (n).     Figure  92  represents  a  crystal  of  calcite 

(CaCO3),  with  the  forms  R  (/>),  5  R  (Y),  4R  (m)  and  the  scaleno- 


hedrons  R3  (r),  R5  (y),  R*  (v),  and  -R3  (/). 

4 


THE   HEXAGONAL   SYSTEM 


75 


Pyramidal  Hemihedrons  (Hexagonal  Bipyramidal  Class).— 
The  pyramidal  hemihedrons  may  be  derived  from  the  holohedrons 
by  extending  the  alternate  pairs  of  planes  meeting  in  the  lateral 
edges.  They  are  characterized  by  the  presence  of  a  principal 
plane  of  symmetry,  an  axis  of  sixfold  symmetry,  and  a  center  of 
symmetry.  The  secondary  planes  of  symmetry  and  the  six  axes 
of  binary  symmetry  of  the  holohedrons  are  lost  (Fig.  93). 

The  only  holohedrons  from  which  new  pyramidal  hemihedrons 
may  be  derived  are  the  dihexagonal  bipyramids  and  prisms. 

The  Pyramids  of  the  Third  Order. — The  dihexagonal 
bipyramid  gives  rise  to  two  bipyramids  of  the  third  order  which 


FIG.  93.  —  Diagram 
illustrating  the  distribu- 
tion  of  the  elements  of 
symmetry  in  pyramidal 
hemihedrons. 


FIG.  94.  —  Positive  hex- 
agonal  bipyramid  of  the 
third  orderj  +  [™^\  or 

ir(hikl}. 


FIG.  95.  —  Negative  hex- 
agonal    bipyramid    of    the 

third   order,  -  f2^]   or  TT 
(hkit). 


are  like  the  pyramids  of  the  first  and  second  orders  in  possessing 
12  similar  triangular  faces  and  a  hexagonal  cross  section  (Figs. 
68  and  70).  They  differ  from  these  bipyramids,  however,  in 
their  relation  to  the  lateral  axes.  In  the  bipyramid  of  the 
third  order  the  axes  terminate  in  the  lateral  edges  somewhere 
between  the  centers  of  the  edges  and  the  solid  angles.  The 


two    forms    are    congruent    and    their    symbols    are    + 

\mPn\  /T  \Pn\       \rnPn] 

+  (Fig.  94),   -     Y  >       ~^~  r    g'  95^         ' 

L    .  ^    "    J  L_1L  J 

indices  are  7t(hikl)  and  n(hkil). 

The  Prisms  of  the  Third  Order. — In  the  same  way  the  dihex- 
agonal prisms  yield  prisms  of  the  third  order  (Fig.  96),  which, 


GEOMETRICAL  .CRYSTALLOGRAPHY 


when  observed  alone,  are  not  distinguishable  by  sight  from  the 
prisms  of  the  first  order.  When  in  combination  with  other  prisms, 
however,  they  are  easily  recognizable  from  the  fact  that  the 
lateral  axes  terminate  neither  in  the  interfacial  edges  between 
contiguous  planes  nor  in  the  centers  of  these  planes.  Their 

symbols  are  ±  and  their  indices  x(hiko)  and  x(hkio). 

The  Relations  of  the  Pyramids  and  Prisms  of  the  Three 
Orders.  —  The  pyramids  of  the  three  orders  are  impossible  to 
distinguish  by  the  eye  alone  when  either 
occurs  singly.  If  the  axial  ratio  is  known, 
a  measurement  of  the  lateral  interfacial 
angle  and  a  calculation  from  this  of  the  ratio 
between  the  distances  at  which  the  pyram- 
idal planes  cut  the  a  and  c  axes  will  deter- 
mine the  question.  If  this  ratio  is  the 
same  as  the  axial  ratio,  or  a  multiple  of  it, 


FIG.     96-Hexagonal     the  pyramid  is  of  the  second  order  (c  = 

prism  of  the  third  order,  r* 

+  ["-?"],  orir(hiko).  tan  /?,  where  OE=i.  Compare  p.  58.) 
If  the  axial  ratio  is  different  from  the  de- 
termined ratio,  the  pyramid  is  of  the  first  or  third  order.  If  it 
is  .866  that  of  the  determined  ratio,  or  some  multiple  of  .866, 
the  pyramid  is  of  the  first  order  (compare  p.  58).  If  the  deter- 
mined ratio  is  not  the  same  as  the  axial  ratio  or  some  multiple  of 
it,  and  is  not  .86,  i  .  7,  or  2  .  5  of  it,  the  pyramid  is  of  the  third  order. 

The  three  orders  of  prisms  are  impossible  to  distinguish 
either  by  the  eye  or  by  measurement  when  they  occur  alone,  even 
if  the  axial  ratio  is  known. 

When  in  combination  there  is  little  difficulty  in  discriminating 
between  the  different  orders.  If  the  order  of  any  form  in  the 
combination  is  known,  it  is  only  necessary  to  locate  the  lateral 
axes  to  determine  the  orders  of  all  the  prisms  and  pyramids  in 
the  combination.  The  relation  of  the  orders  to  the  axes  is 
indicated  in  the  subjoined  figure  (Fig.  97)  in  which  the  heaviest 
line  indicates  the  position  of  the  prism  of  the  third  order. 

The  Trapezohedral  Hemihedrons  (Hexagonal  Trapezohe- 
dral  Class).  —  In  the  hemihedrons  of  this  class  all  the  planes  of 


THE   HEXAGONAL    SYSTEM 


77 


symmetry  and  the  center  of  symmetry  have  disappeared.  The 
forms  therefore  possess  no  pairs  of  parallel  planes.  All  the  axes 
of  symmetry  remain  (Fig.  98). 

-IB* 


/ 
\     / 

T 


FIG.  97. — Diagram  illus- 
trating the  relations  of  the 
prisms and,pyramids  of  the  three 
orders  to  the  hexagonal  axes. 


•**— B^ji 


* 


FIG.  98. — Diagram  illus- 
trating the  distribution  of 
the  elements  of  symmetry  in 
h'exagonal  trapezohedral 
hemihedrons. 


FIG.  99. — Right  hex- 
agonal trapezohedron 
r«E»or  r(hikl}. 


FIG.  100. — Left  hexagonal 
trapezohedron,  lmPn  f  or 
r(kihl}. 


Two  new  forms  are  possible.     These  are  known  as  hexag- 
onal trapezohedrons.     They  are  enantiomorphous  (see  page  83), 

mPn 
and    are   represented    by   the   Naumann    symbols   r  -   — ,    and 


ntPn 


The  r  and  /  signify  right  (Fig.  99)  and  left  (Fig.  100). 


Since  no  crystals  bearing  forms  of  this  class  have  been  ob- 
served, it  is  needless  to  discuss  them  in  this  book. 


GEOMETRICAL   CRYSTALLOGRAPHY 


The  Trigonal  Hemihedrons  (Ditrigonal  Bipyramidal  Class). 

-These  forms  have  never  been  observed  except  on  hemimorphic 

crystals.  The  theoretical  forms  possess  one  principal  and  three 
secondary  planes  of  symmetry,  an  axis  of 
trigonal  symmetry,  and  three  binary  axes 
of  symmetry.  Three  of  the  secondary 
planes  of  symmetry  and  three  of  the  axes 
of  binary  symmetry  belonging  to  the  holo- 
hedrons  have  disappeared,  as  has  also 
the  center  of  symmetry.  The  axis  of 
sixfold  symmetry  has  become  an  axis  of 
threefold  symmetry  (Fig.  101).  As  ob- 
served in  connection  with  hemimorphism, 
the  principal  plane  of  symmetry  and  the 

three  axes  of  binary  symmetry  are  lost  and  the  axis  of  trigonal 

symmetry  becomes  polar. 

The  forms  of  this  class  are  characterized  by  triangular  or 

triangle-like  cross  sections. 

The     Trigonal    Prisms    and    Bipyramids. — These    hemihe- 

drons  may  be  derived  from  the  prisms  and  pyramids  of  the  first 

order  by  the  suppression  of  the  planes  in  alternate  sextants.     The 


FIG.  101. — Diagram  illus- 
trating the  distribution  of 
the  elements  of  symmetry 
in  trigonal  hemihedrons. 


FIG.  102. — Positive  trigonal 

OOP 


FIG.  103. — Negative  trigonal 

oo  P  ,7 

prism, —-^-,   or  ohho. 


results  are  two  congruent  prisms  and  two  congruent  bipyramids 
with  cross  sections  that  are  equilateral  triangles.  Two  lateral 
axes  terminate  in  each  of  the  planes  of  the  prism  and  in  each 
lateral  edge  of  the  pyramid  in  such  a  manner  that  the  distance 
between  the  polar -edges  is  divided  into  three  equal  spaces. 


THE    HEXAGONAL    SYSTEM 


79 


The  new  forms  are  known  as  the  trigonal  prisms  and  bipyra- 

ooP 

mids.     There  are  two  of  each  with  the  symbols  H —   -  (Fig.    102) 

2 

ooP  mP  mP 

and—      -  (Fig.  103),  and+  -   -   (Fig.  104)  and —    -  (Fig.  105). 
222 

Their  corresponding  indices  are  hoho,  ohho  and  kohl  and  ohhl. 


FIG.    104. — Positive     trigonal 
bipyramid  +— ,  or  hohl. 


FIG.  105. — Negative  trigonal 
bipyramid,  —  -^-,  or  ohhl. 


The  Ditrigonal  Prisms  and  Bipyramids. — Similarly  the  di- 
hexagonal  prisms  and  bipyramids  yield  four  new  forms,  two  con- 
gruent ditrigonal  prisms  and  two  congruent  ditrigonal  bipyramids. 
These  possess  two  kinds  of  polar  edges  and  their  cross  sections 


FIG.  1 06. — Diagram  illustrating  relation  of  planes  in  ditrigona   pyramid 
and  prisms  to  the  axes  in  the  hexagonal  system. 

are  six-sided  with  the  sides  arranged  in  pairs  so  as  to  produce 
a  triangle-like  outline.  The  crystallographic  axes  in  the  prisms 
terminate  at  points  between  the  centers  of  the  faces  and  the  blunt 
vertical  edges.  In  the  bipyramids  they  terminate  at  the  corre- 
sponding points  on  the  lateral  edges  (Fig.  106). 


8o 


GEOMETRICAL   CRYSTALLOGRAPHY 


The  symbols  of  the  prism  are  ±—    -  (Figs.  107  and  108)  and 


kOj  ihko,  and  of  the  bipyramids  ±-     -  (Figs.  109  and  no)  and> 

2 

hikl,  ihkl. 


FIG.  107. — Positive  ditrigonal 


FIG.    108. — Negative  ditrig- 


prism,  +  ^-n,  or  hik°. 


Pn 


onal    prism,  —  —  -  ,  or  ihko. 


Combinations  with  Hemimorphism  (Ditrigonal  Pyramidal 
Class). — As  has  already  been  stated,  the  forms  of  this  class  are 
known  only  in  combination  with  hemimorphism.  Consequently 
only  the  upper  or  lower  half  of  each  form  is  present  on  any  one 
crystal.  In  the  case  of  the  prisms,  however,  there  is  no  geometrical 


FIG.  109. — Positive  ditrigonal 
bipyramid,  +  ^-w,  or  hikl. 


FIG.  no. — Negative  ditrigo- 
nal bipyramid,  —  — P— ,  or  ihkl. 


difference  noticeable  between  the  complete  hemihedral  forms  and 
those  that  are  hemimorphic,  since  each  plane  belongs  at  the  same 
time  to  both  upper  and  lower  dodecants.  In  the  case  of 
the  bipyramids,  on  the  other  hand,  half  of  the  planes  exist  either 
at  the  upper  pole  or  at  the  lower  one.  Consequently  there  are 


THE   HEXAGONAL    SYSTEM 


8l 


four  hemimorphic  forms  of  the  bipyramid  possible.  Because  the 
planes  on  the  hemimorphic  pyramids  are  only  1/4  the  total  number 
of  planes  on  the  corresponding  holohedron  their 

mP  niP 

symbols  become  +  --  u,  (hohl),  -\  -----  /    (ohhl), 
4  4 

-jjjT)  •»wT* 

u  (ohhl)  and  -        /  (hohl),  in  which  the 

4  4 

letters  u  and  /  after  the  Naumann  symbols 
indicate  upper  and  lower. 

The  trigonal  hemihedrons  are  easily  recog- 
nized in  combinations  by  their  cross  sections. 

The  best  illustration  of  this  type  of  hemi- 
hedrism  is  shown  by  the  mineral  tourmaline     combinations  of  trig- 

,  ...      J    N       ~  onal  hemihedrons. 

(a  complex  boro-silicate)  ;   figure   in    repre- 


sents  the   combination  of 


(n), 


P  P  2P  OP 

-u(P),  -f-/(P'),  —  -u(o'),  —  /  (c), 
4442 

/  and  -  ™Pl. 


TABULAR   SUMMARY   OF  HEXAGONAL  HEMIHEDRONS. 

(See  explanation  under  list  of  isometric  hemihedrons,  page  51.) 


Hemihedrons 


Holo- 
hedrons 


mPn 

oo  Pw 


00  P2 

mP 

ooP 

oP 


Rhombohedral 

wPw 
± or  mRp 

oo  Pn 
mP2 


mP 

±  —  ,  orR 

2 


±p» 

TooPw 
2 


OP 


mP 

ooP 

oP 


Pyramidal  Trapezohedral      Hem?m°o™L 


mPn 

2 


mP 

ooP 

oP 


mPn 

-- 
4 

oo  Pn 


mP 

±  -—u.l. 
4 
ooP 

2 
OP 


82  GEOMETRICAL   CRYSTALLOGRAPHY 

Combinations  of  Hemihedrons. — The  statements  made 
with  reference  to  the  combinations  of  hemihedrons  in  the  iso- 
metric system  (pp.  51-52)  apply  equally  well  to  those  of  the  hexa- 
gonal and  the  succeeding  systems.  Only  those  hemihedrons  may 
combine  that  have  the  same  grade  of  symmetry;  i.e.,  of  the 
forms  indicated  above,  only  those  may  be  found  in  combination 
that  are  represented  by  the  symbols  in  the  same  vertical  columns. 

TETARTOHEDRAL  DIVISION. 

Tetartohedrism  of  the  Hexagonal  System. — The  tetar- 
tohedral  forms  of  this  system,  like  the  hemihedral  forms,  are  of 
great  theoretical  importance  because  the  most  widely  spread  of 
all  minerals,  quartz  (SiO2),  often  exhibits  well-characterized 
tetartohedral  planes.  Moreover,  they  are  of  considerable  prac- 
tical interest  because  of  the  fact  that  there  is  a  close  relation 
existing  between  certain  important  physical  properties  of  tetar- 
tohedral crystals  and  the  planes  occurring  on  them. 

There  are  three  classes  of  tetartohedrons  in  this  system 
distinguished  from  one  another  by  symmetry.  They  are  known 
as  Trigonal,  Trapezohedral,  and  Rhombohedral  tetartohedrons 
because  their  characteristic  forms  are  trigonal  bipyramids  and 
prisms,  trapezohedrons  and  rhombohedrons.  They  naturally 
possess  a  different  grade  of  symmetry  from  the  hemihedral  forms 
of  the  same  names,  and  therefore  are  designated  as  of  different 
orders  in  the  case  of  the  bipyramids,  prisms,  and  rhombohedrons. 
The  tetartohedral  trapezohedrons  have^three-sided  polar  angles, 
and  therefore  are  distinguished  from  the  corresponding  hemi- 
hedral forms,  which  have  six-sided  polar  angles,  by  prefixing  the 
adjective  trigonal  to  the  name  of  the  form.  Only  the  trapezo- 
hedral  tetartohedrons  will  be  discussed. 

Trapezohedral  Tetartohedrons  (Trigonal  Trapezohedral 
Class). — Only  ten  forms  of  this  class  are  different  geometrically 
from  holohedrons  and  hemihedrons.  Of  these  four  are  trigonal 
trapezohedrons,  two  are  ditrigonal  prisms,  two  are  trigonal 
prisms,  and  two  are  trigonal  pyramids.  The  only  elements  of 
symmetry  in  them  are  an  axis  of  threefold  symmetry  coinciding 
with  the  vertical  crystallographic  axis  and  three  polar  axes 


THE    HEXAGONAL    SYSTEM 


of  binary  symmetry  which  coincide  with  the  lateral  crystallo- 
graphic  axes.  The  loss  of  the  center  of  symmetry  means  the 
absence  of  pairs  of  parallel  planes.  (Fig.  112.) 

The  forms  may  be  regarded  as  being  derived  from  the  holo- 
hedrons  by  applying  to  them  at  the  same  time  the  conditions  of 
rhombohedral  and  trapezohedral  hemihedrism  (Fig.  113);  or 
they  may  be  thought  of  as  being  derived  from  the  rhombohedral 
hemihedrons  by  applying  to  them  the  condition  of  trapezohedral 
he-mi hedrism,  or  vice  versa. 


FIG.  ii2.— Diagram  illus- 
trating distribution  of  sym- 
metry elements  in  trapezohe- 
dral tetartohedrons. 


FIG.  113 — Application  of 
rhombohedral  and  trapezo- 
hedral hemihedrism  to  the 
dihexagonal  bipyramid. 


Trigonal  Trapezohedrons. — The  trigonal  trapezohedrons  con- 
tain six  trapezoid  faces  that  correspond  to  six  faces  of  the  dihex- 


FIG.  114 — Positive 
right  trigonal  trape- 
zohedron,  +  r  *^ 

4       ' 

or  hikl. 


FIG.  115. — Positive 
left  trigonal  trapezo- 
hedron,  +  1™**L }  or 

MM. 


agonal  bipyramid.     They  meet  in  six  equal  polar  edges  and  in 
zig-zag  lateral  edges  made  up  of  long  and  short  lines.     Three 


84 


GEOMETRICAL   CRYSTALLOGRAPHY 


lateral  axes  with  the  same  sign  terminate  in  the  centers  of  the 
long  lateral  edges,  and  the  other  three  in  the  centers  of  the  short 
edges.  Since  each  form  contains  but  one-fourth  the  planes  of 
the  holohedron,  there  are  in  all  four  forms  derivable  from  each 
holohedron  (see  Fig.  113).  These  are  designated  as  positive  ( +), 
negative  (— ),  right  (r),  and  left  (/).  The  +  and  —  forms  are 
congruent  and  the  r  and  /  forms  are  enantiomorphous;  i.e.,  they 
are  symmetrical  with  respect  to  one  another  and  neither  can  be 
so  revolved  that  its  faces  shall  be  parallel  to  the  faces  of  the  other. 

J77  T"^77  /YM  "P1!/ 

The   Naumann    symbols   are    +r —       (Fig.    114),    —  r— :L-, 

4  4 

-f7—      (Fig.  115),  and  —I        .     The  corresponding  indices  are 
4  4 

ihkl,  kihl  and  khil. 


FIG.  116. — Diagram  illustrating  relation  of  planes  to  axes  in  hemihedral  (A) 
and  tetartohedral  (B),  ditrigonal  prisms  and  pyramids. 


\ 


FIG.   117. —  Right  ditrigonal 
prism   of     the    second    order, 


„*£»   or  hiko. 

2 


FIG.     118. — Left     ditrigonal 
prism  of  the  second  order,/0^— "> 

khio. 


The  Ditrigonal  Prisms  of  the  Second  Order. — The  dihexagonal 
prisms  yield  two  ditrigonal  prisms  with  the  symmetry  of  trapezohe- 
dral  tetartohedrism.  These  are  .geometrically  similar  to  the  hemi- 


THE    HEXAGONAL    SYSTEM 


hedral  ditrigonal  prisms.  They  differ  from  them,  however,  in  the 
relation  of  their  planes  to  the  axes  (see  Fig.  116).  Because  of 
this  difference  the  forms  of  this  class  are  known  as  of  the  second 

order,   and  their  symbols  are   written  r—         (Fig.     117)    and 

P  2 

/._       (pjg>  n8),  or  hiko  and  kiho. 


FIG.  119. — Right  trigonal  bi- 
pyramid  of  the  second  order,  r  ™±2} 
or  hh2hl. 


FIG.  120. — Left  trigonal  bipyra- 
mid  of  the  second  order,  /^L2.)  Or 
2hhhl. 


FIG.  121. — Right  trigonal 
prism  of  the  second  order, 
r  *P* ,  or  hhzho. 


FIG.  122. — Left  trigonal 
prism  of  the  second  order, 
/  -  -,  or  2hhho. 


Trigonal  Bipyramids  and  Prisms  of  the  Second  Order. — The 
hexagonal  bipyramids  and  prisms  of  the  second  order  yield 
trigonal  bipyramids  and  prisms  of  the  second  order  which  differ 
from  the  corresponding  hemihedral  trigonal  forms  in  their 
positions  with  respect  to  the  axes — the  relation  of  their  faces 
to  the  axes  being  the  same  as  in  the  hexagonal  pyramid  of  the 
second  order;  viz.,  in  the  prism  each  face  is  perpendicular  to  one 


86 


GEOMETRICAL   CRYSTALLOGRAPHY 


axis,  and  in  the  pyramids  each  lateral  edge  is  at  right  angles  to 
an  axis  (see  Figs.  120  and  122). 

When  in  combination  with  other  forms  the  tetartohedral 
forms  are  easily  recognized  by  their  positions.  To  distinguish 
them  from  other  similar  geometrical  forms  they  are  represented  by 


oo  P2  , 

and  r—   -  and  / 

2 


(Figs,  no,  120,  121,  and  122). 


22 

Combinations.  —  The  mineral  exhibiting  these  forms  in 
greatest  perfection  is  quartz  (SiO2).  Two  crystals  of  this  sub- 
stance are  shown  in  figures  123  and  124.  Crystals  showing  right 
forms  are  referred  to  as  right  crystals,  and  those  showing  left 
forms  are  known  as  left  crystals.  The  former  possess  the  power 
of  rotating  the  plane  of  polarized  light  to  the  right,  and  the  latter 
the  power  of  turning  it  to  the  left.  Since  this  effect  of  quartz 


FIG.  123.  FIG.  124. 

Crystals  of  right-handed  (Fig.  123)  and  left-handed  (Fig.  124)  quartz  crystals 
containing  ooP(a),  +\(r), — j-(z),  +  ^  and  +  r  ^/-*  (s  and  x  in  Fig.  123),  and 
_  ?^_2  and  +  /-^6/s  (S  and  x  in  Fig  I24) 

upon  polarized  light  is  made  use  of  in  the  manufacture  of  certain 
optical  instruments,  the  ability  to  recognize  right  and  left  forms  is 
extremely  valuable.  The  forms  exhibited  by  the  two  crystals 

P  P  2?2  2?2 

figured  are:  <»P(a),  +-(r),  -— (z),  + (s  in  Fig.  123),  - 

222  2 

(s  in  Fig.  124),  +rr-    '5  (x  in  Fig.  123),  +/—  —  (x  in  Fig.  124). 
4  4 


CHAPTER  VIII. 


THE  TETRAGONAL  SYSTEM. 

Similarity  Between  Tetragonal  and  Hexagonal  Systems. 

—Reference  has  already  been  made  (p.  54)  to  the  similarity 
between  tetragonal  and  hexagonal  forms  in  consequence  of  the 
existence  in  them  of  one  principal  plane  of  symmetry  and  a 
number  of  secondary  planes  perpendicular  to  this.  The  dif- 
ferences between  the  two  systems  are  due  to  the  difference  in  the 
number  of  these  secondary  planes  and  in  the  number  of  the  lateral 
axes  to  which  they  give  rise.  We  shall  see,  before  the  discussion 
of  the  tetragonal  system  is  finished,  that  there  is  a  close  analogy 
between  this  system  and  the  hexagonal  system,  and  that  a 
familiarity  with  the  forms  of  the  latter  will  prove  a  great  assistance 
in  obtaining  a  knowledge  of  the  former. 

HOLOHEDRAL  DIVISION. 

(Ditetragonal  Bipyramidal  Class.) 

The  Symmetry  of  the  Tetragonal  System. — The  complete 
forms  of  the  tetragonal  system  are  symmetrical  with  respect  to 
one  principal  plane  of  symmetry  and  four 
secondary  planes  of  symmetry  at  right  angles 
to  this  (Fig.  125).  The  secondary  planes 
intersect  in  a  common  line  and  are  inclined 
to  each  other  at  angles  of  45°.  In  addition 
to  these  planes  of  symmetry  there  are  present 
in  all  holohedrons  in  this  system  also  one  axis 
of  fourfold  symmetry  which  coincides  with  the 
line  of  intersection  of  the  secondary  planes  of 
symmetry,  and  four  axes  of  binary  symmetry 
corresponding  to  the  intersections  of  the 
secondary  planes  of  symmetry  with  the  princi- 
pal plane.  There  is  also  a  center  of  symmetry 
(Fig.  126).  Figure  127  is  a  diagrammatic  cross  section  through  the 
most  general  form  of  the  system,  along  the  plane  of  the  lateral  axes. 

8? 


FIG.  125. — Distri- 
bution of  planes  of 
symmetry  in  tetrag- 
onal holohedrons. 


88 


GEOMETRICAL    CRYSTALLOGRAPHY 


The  Crystallographic  Axes. — The  lines  chosen  as  the  axes  are 
the  lines  of  intersection  of  the  secondary  planes  of  symmetry  and 
the  intersection  of  alternate  secondary  planes  with  the  principal 


FIG.  126.— Model  of 
symmetry  elements  in 
tetragonal  holohedrons. 


FIG.  127. — Distribution 
of  the  elements  of  sym- 
metry of  tetragonal  holo- 
hedrons projected  on  the 
plane  of  the  lateral  axes. 


plane  of  symmetry.  The  axes  (Fig.  128)  are  thus  three  lines  per- 
pendicular to  one  another  at  a  common  point.  The  two  that  lie 
in  the  principal  plane  of  symmetry  are  equivalent;  i.e.,  their 
unities  are  equal  and  their  ends  are  similarly  terminated.  The 


-t-a. 


FIG.  128. — System  of  axes  in 
the  tetragonal  system. 


FIG.   129. — Groundform  of  the 
tetragonal    system.     Symbol    P, 


third  axis  is  different  from  the  other  two — its  unity  is  different 
from  the  unity  on  the  other  two,  and  its  ends,  while  terminated 
alike,  are  terminated  differently  from  the  ends  of  the  lateral  axes. 


THE    TETRAGONAL    SYSTEM  09 

This  axis  "is  the  vertical  axis.  It  is  designated  by  c.  The  two 
similar  axes  are  the  lateral  axes.  They  are  designated  by  a.  The 
symbol  for  the  system  of  axes  is  a  :  a  :  c. 

A  model  representing  the  axes  with  their  unity  lengths  would 
consist  of  three  wires  perpendicular  to  one  another  at  a  common 
point,  one  shorter  or  longer  than  the  other  two  which  would  be 
of  the  same  length. 

The  Groundform  and  the  Axial  Ratio. — The  planes  of  the 
groundform  in  this  system  must  cut  the  two  lateral  axes  at  the 
same  distance  from  their  intersection,  and  the  vertical  axis  at  a 
greater  or  less  distance  than  this  (Fig.  129).  These  distances, 
whatever  they  may  be,  are  taken  as  the  unity  distances  on  the 
several  axes.  The  value  of  the  unity  on  the  vertical  axis  in  terms 


FIG.  130. 


of  that  on  the  lateral  axes  is  the  axial  ratio;  and  this  differs  for 
every  substance  crystallizing  in  the  system.  It  must,  therefore, 
be  calculated  for  each  case,  and  when  once  determined  it  is  ac- 
cepted as  representing  the  ratios  between  the  unities  on  •  the 
vertical  and  the  lateral  axes  on  all  crystals  of  that  substance. 

The  axial  ratio  for  cassiterite  (SnO2)  is  i  :  .6724.  A  plane 
cutting  c  at  .6724  times  the  distance  at  which  it  cuts  a,  cuts 
both  these  axes  at  unity.  The  plane  a  :  a  :  2C  cuts  these  axes  at 
i  :  i  :  1.3448. 


90  GEOMETRICAL    CRYSTALLOGRAPHY 

The  groundform  in  the  system  consists  of  eight  planes,  four 
above  and  four  below  the  principal  plane  of  symmetry.  These 
form  a  bipyramid  whose  symbol  is  P  (Fig.  129),  and  indices  in. 
The  axial  ratio  in  this  system  may  be  determined  in  a  manner 
exactly  analogous  to  that  employed  in  the  hexagonal  system  (see 
pp.  57-58).  A  bipyramid  is  assumed  as  the  groundform  and  the 
angle  at  its  lateral  interfacial  edge  is  measured  (CLD  in  Fig.  130). 
Let  one-half  of  this  equal  /?. 

OX=OL  tan  /?  (i) 
In  the  triangle  OLA  lying  in  the  plane  of  the  lateral  axes 

OA2=OL2+LA2  (2) 

Since  OL  =  LA,  both  being  opposite  to  angles  of  45°,  and  since 
OA  =  unity  on  a,  equation  (2)  becomes 

_i  =  2  OL2 
OL2=i/2,  and 

OL  =  V/i/2  substituting  in  (i)  we  have 
OX  or  c  =  \/i/2  tan  /? 
c  =  .  7 1  x  tan  /? 

The  Ditetragonal  Bipyramid. — The  most  general  form  in  the 
system  is  composed  of  planes  that  cut  the  three  axes  at  different 
distances.     Their  symbols  are  a  :  na  :  me,  etc. 
When  one  of  the  planes  is  present  symmetry 
demands  the  presence  of  fifteen  others,  and 
there  results  a  double  pyramid  consisting  of 
sixteen  faces  that  are  scalene  triangles  (Fig. 
131).      These   meet   in    two   kinds  of   polar 
edges    and    a    series    of    eight    lateral    edges 
that    are    equal.       The   cross   section    is    an 
octagon  with  its  alternate   angles   equal    and 
different    from    the   intervening   ones.      The 
FIG.  131.— Ditetrag-    lateral    solid    angles    are,   therefore,   of    two 
onal  bipyramid,  mPn    kmds,  of  which  the  alternate  ones  are  equal. 

or  rt  k  I* 

The  lateral  axes  terminate  in  the  blunter 
solid  angles  when  n>  2.4142,  and  in  the  more  acute  ones 
when  w<2.4i42. 

It  is  crystallographically  impossible  that  all  the  solid  lateral 
angles  should  be  equal  and  all  the  interfacial  polar  edges  similar, 


THE    TETRAGONAL    SYSTEM  91 

since  in  this  case  the  value  of  the  parameter  n  in  the  general 
formula  a  :  na  :  ma  would  be  the  tan  of  67°  30'  which  is  2 . 4142  +  , 
and  this  is  irrational.  For  let  the  figure  132  represent  a  cross 
section  of  the  bipyramid  through  the  lateral  axes  (OA  and  OD) 
and  let  BC  and  B'C'  be  the  traces  of  the  two  planes  a  :  na  :  me 
and  na  :  a  :  me  in  the  same  octant.  In  the  quadrangle  OBXB' 
the  angles  =  360°.  The  angles  B  +X+B'  =  36o°— the  angle  O 
which  is  90°.  Consequently  the  sum  of  B,  X  and  B'  =  27o°  (i). 
If  we  assume  the  interfacial  angles  to  be  equal,  then  X=B  +  B', 
in  which  B  and  B'  each  equal  half  of  the  interfacial  angle  at 


FIG.  132. 


the  ends  of  the  axes,  measured  in  the  plane  of  the  axes,  and  X 
the  total  value  of  the  interfacial  angle  between  these  two.  Sub- 
stituting the  value  of  X  in  (i),  we  have  26+26'=  270. 
But  B  and  B'  are  equal,  consequently  46  —  270°,  or  6  =  67°  30'. 
The  intercept  of  the  plane  represented  by  BC  is  unity  (OB)  on 
the  axis  OA.  OC  (or  the  intercept  on  the  axis  OD)  =  OB  tan 
67°  30',  or  since  OB  =  i,  OC  =  nat.  tan  of  67°  30'  which  is 
2.4142  +. 

The  symbol  of  the  ditetragonal  bipyramid  is  mPn;  its  indices 
are  hkl. 

The  Ditetragonal  Series. — A  series  of  forms  may  be  derived 
from  the  ditetragonal  bipyramid  in  the  same  way  as  the  dihex- 


GEOMETRICAL   CRYSTALLOGRAPHY 


agonal  series  was  derived  from  the  dihexagonal  bipyramid,  by 
giving  different  values  to  the  parameter  on  c.     Thus : 

(i)  a  :  na  :  me  (3)  a  :  na  :  c 


(2)  a  :  na  :  — c 


m 


(4)  a  :  na  : 


(5)  a  :  na  :  oc 

The  first  three  symbols  refer  to  planes  of  ditetragonal  bi- 
pyramids  that  differ  from  each  other  merely  in  the  distances  at 
which  their  planes  intercept  the  c  axis.  The  fourth  symbol 
represents  a  plane  of  the  ditetragonal  prism  oePw  (Fig.  133),  and 
the  fifth,  the  basal  plane  oP. 

Series  of  the  First  and  Second  Orders. — In  the  symbol 
a  :  na  :  we,  n  may  be  given  different  values,  which  will  affect 
the  character  of  the  form  produced.  If  n  be  given  any  value 


FIG.  133. — Ditetragonal 
prism,  oo Pn  or  hko. 


FIG.   134. — Tetragonal  prism 
of  the  first  order,  ooP  or  no. 


other  than  unity  or  infinity  the  symbol,  becomes  a  :  n'a  :  me, 
which  represents  'a  plane  of  the  ditetragonal  bipyramid  mPnf. 

When  n  =  oo ,  we  have  a  :  oo  a  :  me,  or  mP  oo ,  and  when 
n=i,  we  have  a  :  a  :  me  or  P.  The  first  is  the  symbol  of  a 
bipyramid  of  the  second  order,  and  the  second  that  of  the  first 
order. 

Tetragonal  Bipyramids  and  Prisms  of  the  First  Order.— 
The  symbol  a  :  a  :  me  represents  a  plane  cutting  the  two  lateral 
axes  at  the  same  distance,  which  may  be  taken  as  unity,  and  the  c 
axis  at  some  distance  other  than  the  unity  distance  on  this  axis. 
The  form  mP,  made  up  of  eight  of  the  planes,  which  are  isosceles 


THE    TETRAGONAL    SYSTEM 


93 


triangles,  is  a  bipyramid  of  the  first  order  (Fig.  129)  in  which 
the  lateral  axes  terminate  at  the  solid  angles  formed  by  two  planes 
above  and  two  below  the  principal  plane  of  symmetry.  Its 
indices  are  hhl.  By  changing  the  parameter  on  c  to  unity, 


FIG.  155. — Tetragonal 
bipyramid  of  the  second 
order,  wPoo  or  ho. 


FIG.  136. — Tetragonal 
prism  of  the  second 
order,  ooPoo  or  100. 


infinity,  and  zero,  the  symbols  become  P,  oo  P,  and  oP.  The  first 
is  the  groundform;  the  second,  the  prism  of  the  first  order  (Fig. 
134),  and  the  third,  the  basal  plane.  The  indices  of  the  prism 
of  the  first  order  are  no  and  of  the  basal  plane  ooi. 

Tetragonal  Bipyramids  and  Prisms  of  the  Second  Order. 
—When  n  in  the  symbol  a  :  na  :  me  be- 
comes oo,  the  symbol  becomes  a  :  oca  :  me. 
This  is  a  plane  of  a  bipyramid  which  differs 
from  the  bipyramid  of  the  first  order  only 
in  its  position  with  respect  to  the  axes. 
These  terminate  in  the  centers  of  the  lateral 
edges,  hence,  from  its  analogy  with  the 
hexagonal  bipyramid  of  the  second  order, 
the  form  is  known  as  the  tetragonal  bipyra- 
mid of  the  second  order.  Its  symbol  is 
wPoo  (Fig.  135).  Its  indices  are  hoi. 

From  this  form  the  prism  of  the  second 

order  is  derived  in  the  same  way  that  the  prism  of  the  first  order 
is  derived  from  the  corresponding  pyramid.  Symbol,  <x>Poo. 
Indices,  100.  (Fig.  136.) 


FIG.  137. — Diagram 
illustrating  relations  of 
planes  of  holohedral 
pyramids  and  prisms  to 
axes  in  tetragonal  sys- 
tem. 


94 


GEOMETRICAL   CRYSTALLOGRAPHY 


Relations  between  Pyramids  and  Prisms  of  the  Various 
Orders. — The  relations  existing  between  the  various  pyramids 
and  prisms  of  the  tetragonal  system  are  shown  in  the  diagram, 
figure  137,  which  is  a  cross  section  along  the  principal  plane  of 
symmetry.  The  inner  square  gives  the  position  of  the  bipyramid 
of  the  first  order  with  respect  to  the  lateral  axes,  and  the  outer 


FIG.  138. — Combina- 
tion of  bipyramid  and 
prism  of  same  order. 


FIG.  139. —  Combination 
of  bipyramid  and  prism  of 
different  orders. 


square  that  of  the  bipyramid  of  the  second  order.  The  octagon 
between  these  squares  gives  the  position  of  the  planes  of  the 
ditetragonal  bipyramid. 

Combinations. — The  combinations  of  tetragonal  forms  are 
similar  in  character  to  those  of  hexagonal  forms.  When  the 
axial  ratio  of  a  mineral  is  not  known,  either  of  its  simple  tetragonal 


FIG,  140.  FIG.  141. 

Combinations  of  bipyramids  of  different  orders.  The  pyramid  of  the  second 
order  has  the  shorter  intercept  on  c  in  Fig.  141  and  the  longer  intercept  on  this 
axis  in  Fig.  142. 


prisms  or  bipyramids  may  be  regarded  as  the  form  of  the  first 
order,  and  in  this  way  the  position  of  the  axes  is  fixed.  Other 
simple  bipyramids  and  prisms  belong  to  the  series  of  the  first 
order  or  to  that  of  the  second  order  according  to  their  relations 


THE   TETRAGONAL   SYSTEM  95 

with    these    axes.     Ditetragonal    bipyramids    and    prisms    are 
always  recognizable  by  the  number  of  their  planes. 

Figure  138  is  a  combination  of  a 
bipyramid  and  prism  of  the  same  order, 
and  figure  139  that  of  a  bipyramid  and 
a  prism  of  different  orders.  Figures  140 
and  141  are  combinations  of  pyramids 
of  the  two  orders.  In  the  former  the  ~  , 

FIG.  142. — Crystal  of  an- 

pyramid    of    the    second    order    has    a     atase  containing  1/3  P  (»>, 

,  ,,          ,.  r  ,,  i/7  P(v),  ooP(w),  ooPoo(a) 

shorter  parameter  on  c  than  that  of  the     an^  p «,(«). 

first  order,   and  in  figure  141  it  has  a 

longer  parameter  on  this  axis.     Figure  142  is  a  crystal  of  anatase 

(TiO2)  with  1/3  P(z),  1/7  P(V),  ooP(w),  o>Poo(a),  and  Poo  (e). 

HEMIHEDRAL  DIVISION. 

Hemihedrism  in  the  Tetragonal  System. — Although  the 
hemihedral  division  of  this  system  is  by  no  means  as  important  as 
the  hemihedral  division  of  the  hexagonal  system,  it  nevertheless 
deserves  consideration  since  several  well-known  minerals  exhibit 
forms  belonging  to  it. 

Possible  Kinds  of  Hemihedrism. — In  the  tetragonal  system 
there  are  three  classes  of  hemihedrons  distinguished  by  differences 
in  symmetry.  They  may  be  regarded  as  derived  from  the 
holohedrons  by  the  three  methods  indicated  in  the  subjoined 
figures. 

The  first  method  (Fig.  143)  is  known  as  sphenoidal  hemi- 
hedrism;  the  second,  pyramidal  hemihedrism  (Fig.  144),  and  the 
third,  trapezohedral  hemihedrism  (Fig.  145). 

Forms  derived  by  the  third  method  have  not  been  observed 
on  minerals.  Those  derived  by  the  second  method  are  found  on 
several  rare  minerals.  The  sphenoidal  forms  are  seen  on  the 
common  mineral  chalcopyrite  (CuFeS2). 

The  Sphenoidal  Hemihedrons  (Tetragonal  Scalenohedral 
Class). — This  class  of  hemihedrons  may  be  considered  as  being 
derived  from  the  holohedrons  by  the  suppression  of  the  planes 
in  alternate  octants  and  the  extension  of  all  others.  This  is 


96 


GEOMETRICAL    CRYSTALLOGRAPHY 


analogous  to  tetrahedral  hemihedrism  in  the  isometric  system, 
and  rhombohedral  hemihedrism  in  the  hexagonal  system. 

In  this  class  of  hemihedrons  the  principal  plane  of  symmetry 
and   the   two   alternate  secondary  planes  passing  through   the 


FIG.  144. 


FIG.  145. 


FIG.  143- 

Figures  illustrating  the  possible  methods  of  derivation  of  the  hemihedrons  in 
the  tetragonal  system.  Fig.  143,  sphenoidal;  Fig.  144,  pyramidal;  Fig.  145,  trap- 
ezohedral. 

lateral  axes  disappear,  as  do  also  two  of  the  axes  of  binary  sym- 
metry and  the  center  of  symmetry.  Moreover,  the  fourfold 
axis  is  changed  to  a  binary  axis.  The  remaining  elements  of 
symmetry  are  three  binary  axes  of  symmetry  perpendicular  to  one 

another  and  coinciding  with  the 
crystallographic  axes,  and  the  two 
secondary  planes  of  symmetry  passing 
between  the  lateral  crystal  axes  (Fig. 
146).  Four  new  geometrical  forms 
belong  to  this  class,  two  derived  from 
the  ditetragonal  bipyramid  and  the 
other  two  from  the  bipyramid  of  the 
first  order. 

Tetragonal  Scalenohedrons. — The 
most  general  form  of  the  system, 
mPn,  gives  rise  to  two  congruent 
scalenohedrons,  each  composed  of  8  similar  scalene  triangles 
meeting  in  three  kinds  of  interfacial  angles  and  each  possessing 
two  kinds  of  solid  angles.  The  two  planes  passing  through  the 
edges  in  which  the  vertical  axis  terminates  are  perpendicular  to 


FIG.  146. — Diagram  illus- 
trating distribution  of  symmetry 
elements  in  sphenoidal  hemi- 
hedrons. 


THE   TETRAGONAL   SYSTEM 


97 


each  other.     The  lateral  axes  terminate  in  the  centers  of  the 
edges  which  are  not  in  these  planes. 

mPn  mPn 

The  symbols  of  the  forms  are  + (Fig.  147)   and  -    r— 

(Fig.  148),  or  n(hkl)  and  K(hkl). 


FIG.  147. — Positive  tetragonal 
scalenohedron,   +  *"?-  o  r 

K(hkl). 


FIG.  148. — Negative  tetrag- 
onal scalenohedron,  —  —  n  or 
K(hkl). 


FIG.  149. — Positive 
tetragonal  sphenoid, 
+  ™-or  K(hhl). 


FIG.    150.  —  Negative 
tetragonal  sphenoid, 


FIG.  151. — Crystal  of 
urea  with  oP(c),  ooP(w) 
and  +  -?-  (o). 


Tetragonal  Sphenoids. — The  tetragonal  sphenoids  contain 
half  the  planes  of  the  bipyramid  of  the  first  order.  They  differ 
in  appearance  from  the  isometric  tetrahedron  in  that  they  are 
not  of  equal  dimensions  along  the  three  axes.  The  forms  consist 

7 


98  GEOMETRICAL   CRYSTALLOGRAPHY 

of  four  similar  isoscles  triangles  meeting  in  two  kinds  of  inter- 
facial  edges. 

Their  symbols  are  +          (Fig.  149)  and-      —(Fig.  150),  or 
2  2 

K(hhl)  and  K(hhl).     Figure  151  shows  the  combination  of  oP  (c), 

P 

ooP(w),  and  +-  (o\. 

2 

The  Pyramidal  Hemihedrons  (Tetragonal  Bipyramidal 
Class).  —  The  hemihedrons  of  this  class  may  be  derived  from  the 
holohedrons  by  the  extension  of  all  the 
planes  lying  in  the  alternate  sections  into 
which  the  secondary  planes  of  sym- 
metry divide  them. 

These  forms  retain  the  principal  plane 
of  symmetry,  the  axis  of  fourfold  sym- 
metry, and  the  center  of  symmetry.     The 
FiG.i52.-^iagramilluS-     secondary   planes    and    binary   axes   of 
trating  distribution  of  sym-     symmetry  found  in  the  holohedrons  dis- 

metrv  elements  in  pyramidal  ,„.  N 

hemihedrons.  appear  (Fig.   152). 

There  are  four  new  forms  in  this  class, 

of  which  two  are  derived  from  the  ditetragonal  bipyramid  and 
two  from  the  corresponding  prism.  The  forms  are  congruent. 
They  correspond  to  the  pyramidal  hemihedrons  of  the  hexagonal 
system  and  the  parallel  hemihedrons  of  the  isometric  system. 

Tetragonal  Bipyramids  and  Prisms  of  the  Third  Order.— 
These  are  forms  that  resemble  geometrically  the  bipyramids 
and  prisms  of  the  first  and  second  orders.  They  differ  from 
them,  however,  in  the  relation  of  their  planes  to  the  crystal  axes. 

The  pyramids  of  the  third  order  are  derived  from  the  di- 
tetragonal bipyramids,  hence  their  Naumann  symbols  are 

rwPw~l  [mPnl 

+  I  and  —  -  1  »  the  brackets  indicating  that  they  belong 

L    2    J  L    2    J 

to  the  pyramidal  class.  The  prisms  of  the  third  order  are 
derived  from  the  ditetragonal  prisms,  and  are  represented  by  the 


l 

symbols  +  and  -  -  .     Their  corresponding  indices 

2  2    J 


THE    TETRAGONAL    SYSTEM 


99 


are  x  (hkl)  and  K  (hkl),  and  n  (hko)  and  K  (hko).  The  relations  of 
the  planes  of  the  positive  forms  to  those  of  the  pyramids  and 
prisms  of  the  first,  second,  and  ditetragonal  orders  are  shown  in 
figure  153.  The  heavy  lines  indicate  the  positions  of  the  planes 
belonging  to  the  forms  of  the  third  order. 


FIG.  153.  —  Diagram  illus- 
trating relations  of  planes  of 
pyramids  and  prisms  of  various 
orders  to  the  axes  in  the  tetrag- 
onal system. 


FIG.  154. 
Crystal  of  stol- 
zite  with  P(o)  and 


FIG.  155. — Crystal 
o  f  scheelite  with 
Poo(<0,P(«>),+  [^] 
(*)and[3|3](j). 


Combinations. — In  combination  the  pyramids  and  prisms 
of  the  third  order  are  distinguished  from  those  of  the  first  and 
second  orders  by  their  positions  on  the  crystal.  Figure  154 
represents  a  crystal  of  stolzite  (PbWO4)  bounded  by  P(0)  and 


100 


GEOMETRICAL   CRYSTALLOGRAPHY 


+       -^  (#),  and  ngure  i55>  one  of  scheelite  (CaWOJ  on 
which  occur  P  oo  («),  P  (o),  +  j  ^  I  (h)  and  -    [  ^1  (s). 


YIG.  156.  —  Diagram 
illustrating  distribution  of 
symmetry  elements  in 
trapezohedral  hemihedrons. 


The  Trapezohedral  Hemihedrons  (Tetragonal  Trapezohedral 
Class). — By  the  extension  of  the  alternate  planes  on  the  dihex- 
agonal  bipyramid  two  new  forms  are  derived,  and  these  are 


FIG.  157. — Right  tetrag- 
onal trapezohedron,  r  ^—n 
or  r(hkl). 


FIG.    158. — Left     tetrag- 
onal trapezohedron,  lm^-, 

or  r(hkl}. 


enantiomorphous.  They  retain  the  axis  of  fourfold  symmetry 
and  the  four  axes  of  binary  symmetry,  but  have  lost  all  planes  of 
symmetry  and  the  center  of  symmetry  (Fig.  156).  The  forms 
are  known  as  the  right  and  left  tetragonal  trapezohedrons  and 


THE    TETRAGONAL    SYSTEM 


IOI 


are  given  the  symbols  r         (Fig.  157),  /  —    -   (Fig.  158)  or  T  (hkl) 

and  T  (hkl) .  They  are  characterized  by  having  four  equal  polar 
edges  and  a  zig-zag  lateral  edge  composed  of  long  and  short 
courses.  The  lateral  axes  terminate  in  the  centers  of  the  larger 
zig-zag  edges.  Because  they  have  not  yet  been  found  on  crystals 
they  are  not  discussed  in  detail. 


TABULAR  LIST  OF  HEMIHEDRONS  IN  THE  TETRAGONAL  SYSTEM. 

(See  explanation  under  list  of  isometric  hemihedrons,  p.  51.) 


' 

Hemihedrons 

Holohedrons 

Sphenoidal 

Pyramidal 

Trapezohedral 

mPn 

mPn 

± 

[  mPn  ]                    mPn 

±                                   T.I 

2 

L     2      J 

2 

mPv 

mPvc 

mPv 

mP* 

oo  Pn 

vPn 

±\  °°Pw  1 

oo  Pn 

L    2    J 

ooPoo 

coPco 

ooP  oo 

ooP  oo 

mP 

mP 

mP 

mP 

2 

Hemimorphism. — Hemimorphism  is  observed  in  the  follow- 
ing classes  of  the  tetragonal  system,  viz.,  holohedrons  and 
pyramidal  hemihedrons. 

Tetartohedrism. — Although  tetartohedral  forms  are  possible 
in  this  system,  no  crystals  have  been  observed  with  tetartohedrons 
upon  them,  consequently  they  are  not  discussed. 


CHAPTER  IX. 


THE  ORTHORHOMBIC  SYSTEM. 

Systems  Possessing  No  Principal  Plane  of  Symmetry.— 

As  the  hexagonal  and  the  tetragonal  systems  are  classed  together 
in  consequence  of  the  possession  by  them  of  one  principal  plane  of 
symmetry,  so  the  orthorhombic,  the  monoclinic,  and  the  triclinic 
systems  may  be  united  into  a  group  characterized  by  the  entire 
lack  of  principal  planes  of  symmetry.  These  systems  possess 
certain  analogies  which  are  expressed  in  part  by  the  names  given 
to  their  characteristic  forms.  In  each  system  there  are  three 
axes,  and  these  all  possess  different  unities.  No  two  are  equiva- 
lent, hence  the  symbol  of  the  axes  for  each  system  is  a  :  b  :  c. 

The  Orthorhombic  System. — The  holohedral  division  of  the 
orthorhombic  system  includes  forms  possessing  three  secondary 

planes  of  symmetry.  These  are  per- 
pendicular to  each  other,  and  divide 
space  into  eight  octants  (Fig.  159). 
Its  forms  possess  in  addition  three 
axes  of  binary  symmetry,  perpendic- 
ular to  the  three  planes  of  symmetry, 
and  a  center  of  symmetry  (Figs.  160 
and  161). 

Axes  of  the  System. — The  lines 

chosen  as  the  axes  of  the  system  are  the  lines  of  intersection  of 
the  three  planes  of  symmetry  in  the  holohedrons.  They  are 
consequently  three  lines  at  right  angles  to  each  other  at  a  com- 
mon point  (Fig.  162).  Since  there  is  no  plane  of  symmetry 
situated  between  any  two  of  the  axes,  no  two  of  them  can  be 
equivalent,  hence  no  two  can  possess  the  same  unities.  The 
two  ends  of  each  axis,  however,  are  equivalent  since  they 
are  separated  in  each  case  by  one  of  the  planes  of  symmetry, 
hence,  except  in  cases  of  hemimorphism,  the  two  ends  of  each 

102  \ 


FIG.  159. — Distribution  of 
planes  of  symmetry  in  ortho- 
rhombic  holohedrons. 


THE    ORTHORHOMBIC    SYSTEM 


103 

of  the  center  of 


axis  must  be  similarly  terminated.     Because 
symmetry,  the  forms  must  have  parallel  sides. 

Designation  of  the  Axes. — Since  there  is  no  principal  plane 
of  symmetry,  there  is  no  one  axis  that  differs  from  the  other  axes 
in  any  essential  particular.  Any  one  of  the  axes  may  be  selected  as 
the  vertical  axis,  when  the  other  two  become  the  lateral  axes.  One 
of  these  is  longer  than  the  other.  The  longer  axis  is  (Fig.  162) 
designated  as  the  macroaxis,  and  the  shorter  as  the  brachyaxis. 


FIG.  1 60. — Model  showing  dis- 
tribution of  symmetry  elements  in 
orthorhombic  holohedrons. 


FIG.  161. — Symmetry  ele- 
ments of  orthorhombic  holohe- 
drons, projected  on  the  plane 
of  the  lateral  axes. 


In  the  scheme  of  the  axes,  the  brachyaxis  runs  from  front  to  back 
— is  the  a  axis;  the  macroaxis  from  left  to  right — is  the  b  axis;  and 
the  vertical  axis  from  below  to  above — is  the  c  axis. 

In  the  symbols  representing  the  axes  and  the  crystal  planes, 
the  sign  ~  always  refers  to  the  brachyaxis,  and  the  sign  ~~  to  the 
macroaxis.  The  symbol  of  the  axes  thus  becomes  a  :  b  :  c'. 

Groundform  and  Axial  Ratio. — The  groundform  in  this 
system  is  composed  of  planes  cutting  the  three  axes  at  different 
distances,  which  are  assumed  as  the  unity  (Fig.  163)  distances  on 
the  several  axes.  The  entire  form  consists  of  eight  planes,  one  in 
each  octant,  constituting  a  bipyramid  whose  faces  are  unequi- 
lateral  triangles.  In  cross  section  the  pyramid  is  a  rhomb  (Fig. 
164)  and  not  a  square  (Fig.  165)  as  in  the  case  of  the  tetragonal 
pyramid. 

A  different  groundform  is  chosen  for  each  substance  crystal- 
lizing in  the  system.  From  this  the  relative  lengths  of  the 
unities  on  the  c  and  the  a  axes  are  determined  in  terms  of  the 
unity  on  b,  and  these  are  taken  as  the  unities  to  which  all  of 


104 


GEOMETRICAL   CRYSTALLOGRAPHY 


the  planes  occurring  on  the  crystals  of  this  substance  are  referred. 
The  relative  values  of  the  unities  on  the  three  axes  constitute  the 
axial  ratio,  which  in  this  system  contains  one  more  term  than  the 
axial  ratio  in  the  tetragonal  and  in  the  hexagonal  systems. 


— b- 


FIG.  162. — Axes    of    the 
orthorhombic  system. 


FIG.  163. — Groundform  in  the  or- 
thorhombic system.     P  or  in. 


The  value  of  the  axial  ratio  is  so  characteristic  for  each  mineral 
that  it  is  always  given  under  the  name  of  the  mineral  in  the 
larger  text-books  on  mineralogy.  The  axial  ratio  for  olivine 
[(MgFe)2SiOJ,  for  instance,  is  a  :  b  ^=.4657  :  i  :  .5865.  This 
means  that  the  plane  which  cuts  the  three  axes  at  the  unities 


FIG.  164. 


FIG.  165. 


Cross  sections  of  groundforms  in  orthorhombic  (Fig.  164)  and 
tetragonal  (Fig.  165)  systems. 


intercepts  the  a  axis  at  .4657  times  the  distance  from  the  inter- 
section of  the  axes  as  that  at  which  it  cuts  the  b  axis;  and  the  c 
axis  at  .  5865  times  this  distance.  The  plane  that  cuts  the  three 
axes  at  .9314:  i  :  1.7595,  respectively,  is  the  plane  2a  :  b  :  y. 


THE    ORTHORHOMBIC    SYSTEM  105 

HOLOHEDRAL  DIVISION. 

(Orthorhombic  Bipyramidal  Class.) 

The  Most  General  Form. — The  most  general  form  in  the 
system  is  composed  of  planes  cutting  the  three  axes  at  different 
distances,  which  do  not  bear  to  each  other  the  relations  of  the 
unity  distances.  The  symbol  of  one  of  its  planes  may  be  either 
na  :  b  :  me  or  a  :  nb  :  me.  These  symbols  do  not  represent  dif- 
ferent planes  on  the  same  form,  as  corresponding  symbols  do  in 
the  tetragonal  system,  because  symmetry  does  not  demand  the 
presence  of  both,  when  either  is  present.  The  planes  occur  in- 
dependently. The  plane  na  :  b  :  me  requires  the  presence  of 
na  :  —  b  :  me,  and  also—  na  :  b  :  me  and  —  na  :  —  b  :  me,  above 
the  horizontal  plane  of  symmetry,  and  the  presence  of  four  cor- 
responding planes  below  the  symmetry  plane;  but  none  of  these 
planes  demands  the  presence  of  a  plane  cutting  the  axis  b  at  nb. 

The  plane  a  :  nb  :  me  likewise  demands  the  presence  of  seven 
more  similar  planes,  but  its  presence  on  a  crystal  does  not  demand 
the  presence  of  any  plane  cutting  the  a  axis  at  n.  Consequently 
the  most  general  form  in  the  orthorhombic  system  is  an  eight- 
faced  bipyramid. 

Three  Series  of  Forms. — As  in  the  tetragonal  system  there 
are  three  series  of  pyramids  and  prisms  in  the  orthorhombic  sys- 
tem. These  are  named  with  reference  to  their  relations  to  the 
lateral  axes.  //  is  customary  to  regard  the  shorter  distance  at 
which  a  plane  cuts  the  two  lateral  axes  unity  and  to  call  the  series 
by  the  name  of  the  axis  upon  which  the  parameter  is  not  unity, 
provided  the  two  lateral  axes  are  not  both  cut  at  unity. 

There  are  thus  three  series  of  forms — the  groundform  or 
unit  series,  the  brachy  series,  and  the  macro  series. 

The  Unit  Pyramids  and  Prisms.— The  unit  pyramids  and 
unit  prisms  are  composed  of  planes  cutting  the  two  lateral  axes  at 
unity.  These  two  axes  are  cut  at  different  distances,  but  they  are 
distances  that  bear  to  each  other  the  same  ratio  as  do  the  unities 
on  the  axes.  The  intercept  on  the  vertical  axis  may  be  unity 
or  any  multiple  or  small  fraction  of  this. 

The  unit  pyramids,  called  also  the  orthorhombic  bipyramids, 


IO6  GEOMETRICAL   CRYSTALLOGRAPHY 

are  composed  of  8  unequilateral  triangles  whose  apices  are  at  the 
terminations  of  the  vertical  axis.  A  cross  section  through  the 
lateral  axes  would  exhibit  a  rhomb,  whose  diagonals  would 
represent  these  axes.  The  symbol  of  one  of  its  planes  is  a  :  b  :  me, 
and  the  symbol  of  the  form  is  P  (m)  or  mP  (hhl). 

The  groundform  is  that  member  of  this  series  of  pyramids 
in  which  the  intercept  on  the  c  axis  is  unity.  For  instance,  if  the 
axial  ratio  of  a  given  substance  is  .4657  :  i  :  .5865,  and  its 
crystals  contain  planes  whose  intercepts  on  the  a  and  b  axes  bear 
the  relation  .4657  :  i,  these  planes  belong  to  the  unit  series  of 
pyramids.  If  the  relation  of  the  intercepts  on  the  b  and  c  axes 


FIG.   166. — Groundform  and  two  macro  pyramids  in  the  orthorhombic  system, 

P,  Pn  and  P>?. 

is  as  i  :  .  5865  the  pyramid  is  P,  or  the  groundform.  If  the 
ratio  between  the  intercepts  on  b  and  c  are  as  i  :  1.7595,  tne 
symbol  of  the  form  is  3?  (i.e.,  mP). 

When  m  becomes  oo  the  planes  of  the  forrn  are  all  parallel  to 
c,  and  there  results  the  unit  prism  whose  symbol  is  ooP  or  no. 
This  form  differs  from  the  corresponding  tetragonal  prism  in" 
its  cross  section. 

The  Macroseries. — This  series  consists  of  forms  whose 
planes  cut  the  brachyaxis  (a)  at  unity  and  the  macroaxis  (b)  at 
some  distance  other  than  unity. 

The  most  general  symbol  of  a  plane  belonging  to  this  series 
is  a  :  mb  :  me.  The  form  composed  of  planes  of  this  kind  is  a  macro- 
bipyramid  which  can  be  distinguished  from  the  unit  bipyramid 
only  when  the  unity  on  b  is  known.  Its  symbol  is  mPn  or  (hkl). 


THE    ORTHORHOMBIC    SYSTEM 


107 


The  macro-mark  over  the  n  signifies  that  this  parameter  refers  to 
the  macroaxis.  Fig.  166  represents  the  groundform  and  two 
macropyramids  with  different  intercepts  on  b. 

It  sometimes  happens  that  two  crystallographers  working  on 
the  same  crystal  choose  different  pyramids  for  the  groundform 
and  thus  obtain  different  axial  ratios  for  the  same  substance. 
But  in  these  cases  the  two  groundforms  chosen  may  bear  to  each 
other  the  relations  of  unit  pyramids  to  macropyramids  or_brachy- 
pyramids.  In  the  example  of  olivine 
cited  above  (p.  104)  the  axial  ratio 
accepted  is  .4657  :  i  :  .5865.  The 
groundform  P  cuts  the  three  axes 
at  the  relative  distances  indicated. 
The  plane  a  :  26  :  c,  cuts  these  axes 
at  .4657  :  2  :  .5865:  It  is  easily 
conceivable  that  some  crystal- 
lographer  might  prefer  to  use  as 
the  groundform  the  pyramid  com- 
posed of  planes  cutting  the  three 
axes  at  these  distances.  If  so,  his 
choice  of  axial  ratios  would  be:  .23285  :  i  :  .29325.  The 
original  groundform  would  then  be  a  brachypyramid,  20,  :  b  :  20 
or  2?2.  Thus  in  order  to  distinguish  between  macropyramids, 
brachypyramids,  and  unit  pyramids  on  crystals  it  is :  necessary 
to  know  what  axial  ratio  is  accepted  by  crystallographers-  as  their 
standard  of  reference  for  its  forms. 

When  m  in  the  symbol  mPn  becomes  unity  a  macropyramid 
results  that  differs  from  mPn  in  the  inclination  of  its  faces  to  the 
c  axis.  Its  symbol  is  Pw,  or  (hlh). 

When  m  becomes  oo  we  have  the  form  ooPw,  or  (hko)  which 
is  the  macroprism.  Figure  167  shows  the  relation  between  the 
unit  and  one  of  the  macroprisms. 

The  Brachyseries. — In  addition  to  the  macroseries  there 
is  also  in  this  system  a  series  of  brachypyramids  (Fig.  168)  and 
prisms,  which  differs  from  the  macroseries  in  the  fact  that  the 
lateral  parameter  which  is  not  unity  applies  to  the  brachyaxis. 
The  symbols  of  the  brachyseries  are  distinguished  from  those 


FIG.  167. — The  unit  prism  and 
a  macroprism  in  the  orthorhombic 
system,  oo  P  (no)  and  ooPw  (hko). 


108  GEOMETRICAL   CRYSTALLOGRAPHY 

of  the  macroseries  by  the  use  of  the  mark  over  the  parameter 
that  refers  to  the  brachyaxis.  Pw,  mPn,  (khl),  etc.,  are  brachy- 
pyramids,  while  ooPw  (kho)  is  the  brachy prism. 

Combinations. — In  combination  the  forms  of  the  unit  series, 
the  macroseries,  and  the  brachyseries  are  not  difficult  to  dis- 
tinguish when  they  occur  together,  provided  the  unit  forms  can 
be  recognized.  The  macroforms  occur  at  the  terminations  of  the 
brachyaxis  and  the  brachyforms  at  the  termination  of  the 
macroaxis.  When  the  forms  of  either  series  occur  alone,  how- 


Fio.   1  68.  —  Groundform  and  two  brachy  bipyramids  in  the 
orthorhombic  system,  P,  Pw  and  P«'. 

ever,  they  can  be  recognized  only  by  the  measurement  of  their 
interfacial  angles  and  the  calculation  from  these  of  the  parameters 
on  the  lateral  axes  (see  Fig.  174). 

Domes.  —  There  are  other  classes  of  prisms  in  this  system 
that  have  been  given  the  distinctive  name  domes.  These  consist 
of  planes  that  are  parallel  to  one  of  the  lateral  axes,  while  they 
intercept  the  other  lateral  axis  and  the  vertical  axis  at  certain 
definite  distances.  When  parallel  to  the  macroaxis,  the  forms 
produced  by  them  are  called  macrodomes  (Fig.  169);  when 
parallel  to  the  brachyaxis,  the  forms  are  the  brachy  domes  (Fig. 


In  writing  the  symbols  of  the  domes  the  parameter  on  the 
lateral  axis  which  is  not  <x>  is  made  unity.  Thus  the  symbols  of 
planes  of  the  macrodomes  are  a  :  oo£  '<  :  c,  or  a  :  006:  me,  accord- 
ing as  the  c  axis  is  cut  at  unity  or  at  some  other  distance.  When 
m  becomes  <x>  ,  the  plane  is  called  a  pinacoid.  (See  next  section.) 


THE    ORTHORHOMBIC    SYSTEM 


I09 


The  symbols  of  the  macrodomes  are  P  oo,  mP  oo,  (hoi),  and  of 
the  brachy domes,  P  <»,  mP  oo,  (ohl).  Each  dome  consists  of  four 
faces,  uniting  in  pairs  at  the  terminations  of  the  vertical  axis. 
The  macrodomes  are  in  the  angles  between  the  vertical  and  the 
brachyaxis  (Fig.  169),  and  the  brachydomes  in  the  angles  between 
the  vertical  and  the  macroaxis  (Fig.  170). 

Pinacoids. — The  pinacoids  embrace  those  forms  with  planes 
parallel  to  two  axes  at  the  same  time.  Each  form  consists  of  a 
pair  of  parallel  planes.  When  these  are  parallel  to  the  two  lateral 


FIG.  169.  —  Two  ortho- 
rhombic  macrodomes,  P  oo 
and  wPcc  ,  101  and  hoi. 


FiG.  170. — Two  ortho- 
rhombic  brachydomes,  P»o  and 
wPco  or  on  and  ohl. 


axes  they  constitute  the  basal  plane  or  the  basal  pinacoid,  similar 
to  the  corresponding  form  in  the  tetragonal  system.  Its  symbol 
is  written  oP  or  (ooi). 

When  the  planes  of  the  fo'rm  are  parallel  to  the  vertical  and 
the  macroaxis,  the  form  is  termed  the  macropinacoid,  and  when 
parallel  to  the  vertical  and  the  brachyaxis  it  is  called  the  brachy- 
pinacoid.  The  symbol  of  the  former  is  ocPoo(ioo),  and  of  the 
latterooPoo,  or  oio  (Fig.  171). 

Closed  Forms. — It  will  be  noted  that  the  only  forms  that  will 
completely  enclose  space  in  the  orthorhombic  system  are  the 
bipyramids.  No  prism,  pinacoid,  or  dome  will  alone  enclose  space. 
All  are  open  forms.  Consequently  any  crystal  that  contains  on 
it  a  plane  belonging  to  any  one  of  these  three  forms  cannot  be 
represented  correctly  by  less  than  .two  symbols,  no  matter  how 
simple  the  habit  of  the  crystal. 

Combinations. — From    a    consideration    of   the   statements 


no 


GEOMETRICAL   CRYSTALLOGRAPHY 


made  in  the  preceding  paragraph,  it  is  plain  that  the  number  of 
forms  present  in  combination  on  orthorhombic  crystals  is  usually 
larger  than  the  number  found  on  tetragonal  crystals.  Figure  172 
represents  a  combination  of  ocP(/>),ooP  <»(&),  oP(c),  and  2?  £>($, 
while  figure  173  illustrates  a  more  complicated  combination  of 
*P(£),  «>P3(/a  oP(c),  P<2(g),  P(o),  i/2P(V),  i/3P(0")»  and 
2/3?2  (#) .  Figure  1 74  represents  a  combination  sometimes  seen  on 
andalusite  (Al2SiO5),  ooP^(a),  <xpo>(&),  oP(c),  ooP(w),  ooP2(/), 
,  Poo(r),  P(/>),  Poo (^),  and  2  P2(&).  A  very  complex 


^^                 OP 

^^| 

1 

^'"1 
col}* 

cop  as 

1 

x>|Xf            r    " 

x^ 

FIG.  172.^ — Crystal  of  olivine  with 


FIG.  171. — Combination  of 
orthorhombic  pinacoids  and 
basal  plane. 


crystal  of  olivine  [(FeMg)  SiOj  is  shown  in  Fig.  175.  It  contains 
the  forms  ooP^  (a),  oP  (c),  ooP  (n),  ooP2  (s),  ooP3  (r),  Poo(^), 
P^  (h),  2P^  (k),  4P^  W,  P  W,  i/2P  (o),  2P2  (/),  and 


Orthorhombic  crystals  are  frequently  elongated  in  the  direc- 
tion of  one  axis.  This  direction  is  usually  taken  as  the  vertical 
axis  of  the  crystals,  and  the  latter  are  said  to  possess  a  prismatic, 
a  columnar,  or  an  acicular  habit  (Fig.  172).  When  dispropor- 
tionately shortened  in  the  direction  of  a  single  axis,  this  axis  is 
likewise  often  regarded  as  a  vertical  axis,  and  the  crystal  is  said 
to  be  tabular  in  habit. 

Hemimorphism  (Orthorhombic  Pyramidal  Class).  —  The  hemi- 
morphic  development  of  forms  in  this  system  is  of  considerable 
importance.  With  the  disappearance  of  the  planes  of  the  holo- 
hedrons  from  one  end  of  an  axis  there  is  necessarily  the  disap- 


THE    ORTHORHOMBIC    SYSTEM 


III 


pearance  of  one  of  the  planes  of  symmetry,  the  two  axes  of  sym- 
metry lying  in  this  plane,  and  the  center  of  symmetry.  The  ele- 
ments of  symmetry  remaining  are  two  planes  of  symmetry  and 
the  axis  of  binary  symmetry  coinciding  with  their  intersection. 


FIG.  173. — Crystal  of 
topaz.  See  text  for 
symbols  of  planes. 


FiG.  174. — Crystal  of 
andalusite.  See  text  for 
symbols  of  planes. 


This  axis  is  necessarily  polar,  because  of  the  absence  of  the  plane 
of  symmetry  perpendicular  to  it.  It  is  usually  made  the  vertical 
axis  c. 

All  forms  except  those  parallel  to  two  axes,  i.e.,  the  prism,  the 


FIG.  175. — Crystal  of  olivine.     See  text  for  symbols  of  planes. 

macropinacoid,  and  the  brachypinacoids,  now  separate  into  upper 
and  lower  halves,  either  of  which  may  occur  alone. 

Figure   176  represents  a  crystal  of  struvite  (NH4MgPO4  + 

Poo  P  oo 

6H2O)  containing ooP  oo(7>);  at  the  upper  pole,      -u(r),  -      u(q), 


112  GEOMETRICAL   CRYSTALLOGRAPHY 

4?  oo  oP  j./  \x  ^ 

w  (0');  and  at  the  lower  pole,         /  (c)   and  -  /    (rf). 

2  22 

Figure  177  is  a  crystal  of  calamine  (Zn2(OH)2SiO3)  with  ooP  oo  (6), 

7?  00  POO 

oo  Poo  (a),    ooP(£);  at  the  upper  pole,  w         ^   (/),  -       u    (r), 

2  2 

^P  00  P  00  oP  2?  2 

—  u  (q') ,  —  —  «  (0) ,       w  (c) ;  and  at  the  lower  pole, /  (0) . 

222  2 


FIG.  176. — Hemimorphic 
crystal  of  struvite.  See  text 
for  symbols  of  planes. 


FIG.  177. —  Hemi- 
morphic crystal  of  cala- 
mine. See  text  for  sym- 
bols of  planes. 


HEMIHEDRAL  DIVISION. 

Kinds  of  Hemihedrism.  —  The  only  forms  in  the  ortho- 
rhombic  system  that  can  yield  new  hemihedrons  are  the  pyramids. 
One-half  the  faces  of  these  forms  may  be  extended  in  three  dif- 
ferent ways,  but  only  in  one  case  will  new  forms  be  derived  which 
will  comply  with  the  demand  of  hemihedrism.  Consequently 
there  is  in  this  system  but  one  kind  of  hemihedrism,  and  by  it  only 
one  new  type  of  hemihedral  form  is  produced. 

Sphenoidal  Hemihedrism  (Orthorhombic  Bisphenoidal  Class)  . 
—The  only  hemihedrons  in  this  system  are  two  sphenoids  which 
may  be  regarded  as  derived  from  the  various  pyramids  by  the 
extension  of  planes  in  alternate  octants.  These  new  forms 
possess  only  three  axes  of  binary  symmetry.  The  three  planes  of 
symmetry  and  the  center  of  symmetry  have  disappeared  (Fig. 


This  type  of  hemihedrism  is  analogous  to  the  inclined  hemi- 
hedrism of  the  isometric  system  and  the  sphenoidal  hemihedrism 
of  the  tetragonal  system. 


THE    ORTHORHOMBIC    SYSTEM  113 

Sphenoids. — The  sphenoids  are  four-sided  closed  figures  com- 
posed of  four  scalene  triangles  meeting  in  six  interfacial  edges, 
two  of  which  are  equal  and  are  different  from  the  other  four  which 
are  also  equal.  The  vertical  axis  terminates  in  the  centers  of  the 
two  equal  edges,  and  the  lateral  axes  in  the  centers  of  the  other 


FIG.  178. —  Diagram  ill  us 
1  rating  distribution  of  sym- 
metry elements  in  orthorhombic 
sphenoidal  hemihedrons. 


FIG.  179. — Right  orth- 
orhombicbi  sphenoid 
r™P,  or  hhl. 


FIG.  180. — Left  ortho- 
rhombic    bisphenoid, 


four.  The  form  differs  in  appearance  from  the  tetragonal  sphen- 
oid from  the  fact  that  the  edges  at  the  terminations  of  the  vertical 
axes  are  not  at  right  angles.  The  difference  between  the  value 
of  this  angle  and  90°  increases  with  the  difference  in  the  lengths 
of  the  two  lateral  axes. 

From  every  pyramid  two  of  these  forms  are  produced.     They 
are  enantiomorphous,  and  are  therefore  known  as  right  (Fig.  179) 
8 


GEOMETRICAL   CRYSTALLOGRAPHY 


and  left  forms  (Fig.  180).     Their  symbols  are  r  —  (hhl),    I  —  , 

2  2 


r—  (kht),  I—  (hkl), 


-(hkl),  and  £**  (khl). 


FIG.  181—  Crystal 
of   epsom  salts  with 
and  r?o). 


A  combination  of  ooP(/>)  and  r     (0)  as  seen  on  crystals  of 


epsom  salts  (MgSO4  +7H2O)  is  illustrated  in  Fig.  181. 


CHAPTER  X. 


THE  MONOCLINIC  SYSTEM. 

Symmetry  of  the  Holohedrons  of  the  Monoclinic  System. 

—Complete  forms  belonging  to  the  monoclinic  system  possess 
but  a  single  plane  of  symmetry.  Consequently  monoclinic, 
holohedrons  are  bilaterally  symmetrical;  i.e.,  they  possess  two 
sides  that  are  alike.  They  possess  also  an  axis  of  binary  sym- 
metry perpendicular  to  the  plane  of  symmetry  and  a  center  of 
symmetry  (Fig.  182).  Figure  183  is  a 
diagrammatic  representation  of  the  dis- 
tribution of  the  elements  of  symmetry  in 
the  plane  of  the  lateral  axes. 


FIG.  182.— Model  of 
monoclinic  crystal  show- 
ing distribution  of  ele- 
ments of  symmetry  in 
holohedrons. 


FIG  183. — Diagram  illus- 
trating distribution  of  sym- 
metry elements  of  monoclinic 
holohedrons. 


Axes. — The  single  plane  of  symmetry  in  this  system  deter- 
mines the  position  of  one  line  which  may  be  chosen  as  one  of  the 
axes  of  reference  for  the  planes  of  monoclinic  forms.  This 
line  is  the  axis  of  symmetry  which  is  perpendicular  to  the  plane 
of  symmetry.  The  other  two  axes  must  necessarily  lie  in  the 
plane  of  symmetry,  but  their  directions  in  this  plane  are  a  matter 
of  choice,  to  be  decided  in  the  case  of  each  substance  as  may 
be  most  convenient. 

The  two  axes  that  lie  in  the  plane  of  symmetry  are  at  right 
angles  to  the  third  axis,  but  are  inclined  to  each  other  at  some  angle 


Il6  GEOMETRICAL   CRYSTALLOGRAPHY 

other  than  90°.     Their  angle  of  inclination  is  always  designated 
as  the  angle  /?  (Fig.  184). 

Symmetry  demands  that  the  two  terminations  of  the  axis 
that  is  normal  to  the  plane  of  symmetry  shall  be  equivalent  in  all 
respects.  There  is,  however,  nothing  in  the  symmetry  of  the 
system  which  necessitates  the  equivalency  of  any  two  of  the  three 

axes  or  of  the  opposite  ends  of  the 
axes  in  the  plane  of  symmetry. 
Hence  the  unities  on  the  three  axes 
are  different  in  value. 

A  model  representing  the  axes 
of  the  monoclinic  system  would  be 
composed  of  three  lines  of  unequal 
lengths  intersecting  at  a  common 
point.  Two  of  these  must  neces- 
sarily be  inclined  to  each  other  at 
FIG.  184.— Relation  of  axes  to  some  angle  other  than  90°,  and  the 

plane  of  symmetry  in  monoclinic      third   must   be    norma}  to  the  plane 
holohedrons. 

of  these. 

Designation  of  the  Axes. — It  is  customary  in  studying 
monoclinic  crystals  to  place  them  in  such  a  position  that  the  plane 
of  symmetry  shall  stand  vertically.  The  line  normal  to  this  plane 
then  takes  the  position  of  the  b  axis.  Of  the  other  two  axes,  one 
is  made  vertical  and  the  other  is  so  placed  that  it  inclines  toward 
the  observer.  The  acute  angle  /?  is  thus  on  the  back  of  the 
crystal  (Fig.  184). 

As  in  the  other  systems,  the  axis  that  stands  vertically  is  the 
axis  c  or  the  vertical  axis.  The  axis  corresponding  to  the  b 
axis  of  the  orthorhombic  system,  i.e.,  the  one  that  is  normal  to 
the  plane  of  symmetry,  is  the  orthoaxis.  The  third  axis — the 
one  inclined  to  the  vertical  axis — is  the  a  axis.  It  is  known  as 
the  clinoaxis.  The  sign  used  in  symbols  to  designate  reference 
to  the  b  axis  is  the  same  as  that  which  designates  reference  to  the 
macroaxis  in  the  orthorhombic  system.  The  sign  used  to 
designate  the  clinoaxis  is  \.  The  symbol  for  the  axes  is  thus 
a  :  b  :  c. 

Groundform  and  Crystallographic  Constants. — As  in  the 


THE    MONOCLINIC    SYSTEM  117 

orthorhombic  system,  so  in  the  monoclinic  system,  any  form 
composed  of  planes  cutting  the  three  axes  at  finite  distances  may 
be  assumed  as  the  groundform,  and  from  it  the  lengths  of  the 
unities  on  the  three  axes  may  be  determined.  These  unities  are 
expressed  in  terms  of  the  unity  on  b  as  i,  and  when  once  deter- 
mined  for  a  crystal  of  any  substance  this  ratio  is  accepted  as  the 
axial  ratio  for  all  crystals  of  that  substance. 

In  addition  to  the  axial  ratio  there  is  one  other  determination 
necessary  to  fix  the  position  of  the  axes  in  this  system,  viz.,  the 
inclination  of  the  a  to  the  c  axis,  or  the  value  of  the  angle  /?. 

The  axial  ratio  and  the  value  of  /3  constitute  the  crystal- 
lographic  constants.  In  orthoclase  (KAlSi3O8)  these  constants 
are  a  :  b  :  ^  =  .6585  :  i  :  .5554.  £=63°  56' 46",  and  in  augite 
(a  calcium,  magnesium,  iron  silicate)  a  :  b  :  c=  1.092 13  :  i  : 
.58931.  0=74°  10' 9". 

In  practice,  although  the  choice  of  the  lines  that  shall  serve 
as  the  a  and  the  c  axes  is  purely  arbitrary,  it  is  usual  to  select  those 
that  will  yield  the  simplest  symbols  for  the  forms  most  fre- 
quently found  in  combination.  One  or  the  other  of  these  axes  is 
made  parallel  to  some  prominent  plane  on  the  crystal  or,  better, 
to  some  prominent  zone  of  planes.  These  planes  then  become 
pinacoids,  domes,  or  prisms,  when  they  are  easy  to  recognize. 

• 
HOLOHEDRAL  DIVISION. 

(Prismatic  Class.) 

Pyramids. — The  most  general  symbol  possible  in  the  system 
is  na  :  b  :  me,  or  a  :  nb  :  c.  Either  represents  a  plane  cutting 
the  three  axes  at  different  distances,  one  of  which,  the  intercept 
on  b  or  on  a,  is  considered  as  the  unity  on  this  axis.  The  presence 
of  one  of  these  planes  necessitates  the  presence  of  a  corresponding 
plane  on  the  opposite  side  of  the  plane  of  symmetry,  and  the 
presence  of  these  two  demands  the  presence  of  two  others  parallel 
to  the  former  because  of  the  existence  in  the  system  of  a  center  of 
symmetry.  Consequently,  in  this  system  the  most  general  form 
consists  of  two  pairs  of  planes.  If  one  of  the  planes  occurs  in 
an  octant  containing  the  acute  angle  /?,  all  other  planes  of  the 


n8 


GEOMETRICAL   CRYSTALLOGRAPHY 


form  occur  also  in  acute  octants.  These  are  known  as  the 
positive  forms.  If,  on  the  other  hand,  the  planes  are  all  in  the 
obtuse  octants,  negative  forms  result. 

A  combination  of  a  positive  and  a  negative  form  whose 
planes  possess  the  general  symbol  na  :  b  :  me,  gives  a  bipyramid 
(Fig.  185)  analogous  to  the  bipyramids  in  the  orthorhombic 
system,  except  for  the  fact  that  it  consists  of  two  sets  of  planes  of 
different  shapes.  Since  the  positive  and  the  negative  forms  each 
constitute  half  of  this  bipyramid,  they  are  termed  the  positive  and 
the  negative  hemipyramids.  A  monoclinic  bipyramid  com- 
posed of  the  planes  a  :  b  :  c,  must  thus  be  represented  by  two 


FIG.  185. — Combina- 
tion of  positive  and 
negative  hemipyramids. 


FIG.  186. — Crystal  of 
ferrous  sulphate. 


symbols:  +P  and  —P.  Either  of  these  hemipyramids  may 
occur  independently  of  the  other,  as  each  is  completely  holohedral. 
(See  Figs.  186  and  188,  p.  120.) 

Of  these  hemipyramids  there  are  three  different  kinds  corre- 
sponding to  the  three  kinds  in  the  orthorhombic  system.  They 
are  named  in  accordance  with  the  same  principles  that  determine 
the  naming  of  the  orthorhombic  forms. 

a  :    b  :  c  =  +P(iu)  and— P(in),  unit  hemipyramids. 
a  :  nb  :  c=  -fPn  and— Pn,  orthohemipyramids. 

na  :    b  :  c=  +Pn  and—  Pn,  clinohemipyramids. 
When  the  parameter  on  the  vertical  axis  becomes  m  there  result 
other  pyramids   that   belong   to  one  of   these   three  series,   as 
±mP(hhl),  ±mPn(hkl,h>k)  and  ±mPn(M)h<k). 

When  occurring  alone  the  three  hemipyramids  can  be  dis- 


THE   MONOCLINIC   SYSTEM  IIQ 

tinguished  from  each  other  only  by  a  comparison  of  their  inter- 
facial  angles  with  the  corresponding  angles  on  the  assumed  ground- 
form.  When  in  combination  the  orthohemipyramids  and  the  clino- 
hemipyramids  may  often  be  distinguished  from  each  other  by  the 
fact  that  the  orthoforms  occur  near  the  terminations  of  the  clino- 
axis,  while  the  clinoforms  occur  near  the  terminations  of  the 
orthoaxis. 

Prisms. — When  the  intercept  m  in  the  symbols  mP,  mPn,  and 
mPn  is  made  oo ,  the  pyramids  which  they  represent  become  prisms, 
and  we  have  the  unit  prism,  the  orthoprism,  and  the  dino prism. 

Whether  the  prismatic  faces  are  regarded  as  derived  from 
the  positive  or  from  the  negative  hemipyramids,  an  inspection  of 
figure  185  will  show  tnat  each  of  their  planes  must  be  at  the  same 
time  in  both  a  positive  and  a  negative  octant.  Hence  the  use  of 
signs  in  the  symbols  of  the  prisms  is  not  necessary.  QO  P  ( 1 10)  is  the 
unit  prism,  <x)Pn(hko,  h>k),  the  orthoprism  and  <x>Pn(kho,  h<k), 
the  clinoprism.  (See  Figs.  188  and  189.) 

Domes. — The  domes  in  the  monoclinic  system,  like  those  in 
the  orthorhombic  system,  are  composed  of  planes  that  are  parallel 
to  one  of  the  lateral  axes. 

The  orthodomes  are  parallel  to  the  ortKoaxis.  There  are 
two  series:  the  positive  orthodomes,  POQ(/K>/),  mP<x>(hol),  etc.; 
and  the  negative  orthodomes,  — P  oo,  —  mP  GO,  etc.,  the  former  in 
the  acute  /?  and  the  latter  in  the  obtuse  ft  (see  planes  r  and  r'  in 
Fig.  1 86). 

The  clinodomes  are  parallel  to  the  clinoaxis.  These  are  at 
the  same  time  in  both  positive  and  negative  octants,  and  hence 
possess  no  signs.  Their  symbols  are  Poo(oW),  mP&(ohl),  etc. 
(Plane  q  in  Fig.  186). 

Pinacoids. — The  pinacoids  are  parallel  to  two  of  the  axes 
at  the  same  time.  Their  planes  must  belong  to  two  octants,  hence 
they  possess  no  signs. 

The  basal  pinacoid  corresponds  to  the  basal  planes  of  other 
systems.  Its  symbol  is  the  conventional  oP(ooi). 

The  other  two  pinacoids  are  the  orthopinacoid  and  the  clino- 
pinacoid.  The  former  is  parallel  to  the  vertical  and  the  ortho- 
axis,  and  the  latter  to  the  vertical  and  the  clinoaxis.  Their 


120 


GEOMETRICAL    CRYSTALLOGRAPHY 


symbols  are,  respectively,  ooP  00(100)  and  ooP  00(0 10).  Theclino- 
pinacoid  is  the  plane  of  symmetry  (plane  b  in  Fig.  186  and  M 
in  Figs.  187  and  188). 

Combinations. — Combinations  of  monoclinic  forms  are  more 
varied  than  are  the  combinations  of  the  forms  belonging  to  any 
system  of  a  lower  grade  of  symmetry.  This  is  due  to  the  fact  that 
there  are  no  closed  forms  in  this  system.  The  simplest  crystals 
must  be  represented  by  at  least  two  symbols. 

In  working  out  the  symbols  of  monoclinic  crystals  it  is  abso- 
lutely necessary  first  to  bring  the  crystals  into  the  correct  conven- 
tional position.  The  plane  of  symmetry  must  first  be  discovered, 
and  then  must  be  placed  vertically  and  at  the  same  time  parallel 


FIG.  187.  FIG.  188. 

Crystals  of  orthoclase  withooPoo(M),oP(P),  ooP(T),2P5o  («),  +  2P~o%), 

and  +  P(o). 


to  the  line  of  sight  of  the  observer.  When  held  in  this  position,  the 
b  axis  runs  horizontally  from  right  to  left.  Its  position  is  fixed  by 
the  symmetry  of  the  system.  The  choice  of  the  other  two  axes  is 
a  matter  of  convenience  in  each  particular  case  and  is  regulated 
largely  by  the  habit  of  the  crystal.  If  this  possesses  planes  that 
may  be  regarded  as  pinacoids,  these  planes  serve  to  fix  the  posi- 
tions of  the  two  axes  in  question.  The  planes  assumed  as  basal 
planes  determine  the  position  of  the  clinoaxis. 

The  following  illustration  taken  from  Dr.  Williams's  Elements 

of  Crystallography  may  serve  to  make  these  points  clear.     On  the 

crystal  of  iron-  vitriol  (FeSO4  +7H2O),  represented  in  figure  186, 

the  only  plane  whose  value  is  absolutely  fixed  is  the  plane  of 

„  symmetry,  or  clinopinacoid  (b).     It  is  customary  to  make  c  the 


THE    MONOCLINIC    SYSTEM 


121 


basal  pinacoid  and  p  the  fundamental  prism,  whence  q  becomes  a 
clinodome,  o  a  negative  hemipyramid,  and  r'  and  r  plus  and 
minus  hemiorthodomes.  We  might,  however,  turn  the  crystal 
so  as  to  make  c  the  orthopinacoid,  q  the  prism,  and  p  a  clinodome; 
or  we  might  even  make  rf  the  basal  pinacoid,  and  r  the  orthopina- 
coid, when  o  would  become  the  prism,  c  a  hemiorthodome,  and 
p  and  q  both  pyramids. 

Figures  187  and  188  represent  two  common  combinations  in 
this  system.     The  two  crystals  are  very  different  in  habit  though 


FIG.  189. — Crystal  of 
real  gas.  See  text  for 
symbols  of  forms. 


FIG.  190. — Crystal 
of  gypsum,  with  ooP 
(/>),ooPoo(ft)  and 


they  possess  nearly  the  same  forms.  Both  are  crystals  of  the 
feldspar  known  as  orthoclase  (KAlSi3O8).  In  the  figures, 
M=  ocP^c,  P  =  oP,  T=  ooP,  y=  +  2P<x>,  w  =  2?oo,  x=  +P<*>  and 
o=  +P.  Figure  189  is  a  crystal  of  realgar  (As2S2)  wth  oP(P), 
ooPoo(r),  ocP(Jlf),  ooP2(/),  Poo(»)  and  +P(s).  Figure  190  is 
a  crystal  of  gypsum  (CaSO4+2H2O)  with  ocP(/>),  ooPoo(&)  and 


HEMIHEDRAL  DIVISION. 

(Domatic  Class.) 

Hemihedrism  in  the  Monoclinic  System.—  By  the  selection 
of  half  the  planes  of  a  hemipyramid  in  such  a  way  that  opposite 
ends  of  the  ortho-axis  are  equivalently  terminated,  i.e.,  if  each  end 
is  terminated  by  one  plane,  the  conditions  of  hemihedrism  for 


122 


GEOMETRICAL   CRYSTALLOGRAPHY 


this  system  are  complied  with.  The  resulting  hemihedron  still 
retains  the  plane  of  symmetry,  but  the  axis  of  symmetry  and  the 
center  of  symmetry  are  gone  (Fig.  191). 

The  new  form  derived  from  ±  P  consists  of  the  upper  or  the 
lower  half  of  the  hemipyramid,  and  is  known  as  the  tetrapyramid. 
Each  hemipyramid,  therefore,  yields  two  hemihedrons,  which 
are  designated  the  upper  and  the  lower  tetrapyramids.  Their  sym- 

P        P7      P        P7      mP  mPn          mPn 

bolsareH — u,  H — /, — u, — 1,± u.l,±      —u.l,±        u.L,  etc. 

22222  2  2 


FIG.  191. — Diagram  illus- 
trating distribution  of  sym- 
metry elements  in  monoclinic 
hemihedrons. 


FIG.  192. — Crystal 
cf  K2S4Oe  showing 
hemihedral  forms. 
See  text  for  symbols 
of  forms. 


In  the  same  way  the  orthodomes  yield  upper  and  lower 
tetraorthodomes,  the  clinodomes  yield  upper  and  lower  hemidino- 
domes,  and  the  prisms  front  and  rear  hemiprisms.  The  basal 
plane  yields  an  upper  and  a  lower  form,  and  the  orthopinacoid 
a  front  and  a  rear  form. 

oP  co  P  GO  ooPoo 

Figure   192  is  a  combination  of  — u  (c),  j  (a), r 

22  2 

ooP  ooP  Poo  P2 

(a  behind), f(m)  and r  (m  behind) ,          u  (q), —  /  (u),  and 

22  22 

p 

H — l(o),  on  crystals  of  potassium  tetrathionate  (K2S4O6). 

Hemimorphism  (Sphenoidal  Class). — If  one-half  the  planes 
of  the  hemipyramid  are  grouped  at  one  end  of  the  orthoaxis,  the 
other  half  being  absent,  the  plane  of  symmetry  disappears,  the 


THE    MONOCLINIC    SYSTEM 


123 


axis  of  symmetry  becomes  polar,  and  the  center  of  symmetry  no 
longer  exists.  The  resulting  form  has  but  one  element  of  sym- 
metry which  is  the  polar  axis  of  binary  symmetry  (Fig.  193)  and 
it  is  hemimorphic  along  the  orthoaxis. 


FIG.  193. — Diagram  illstrating  symmetry  elements 
in  monoclinic  hemimorphs. 


Each  hemipyramid  thus  breaks  up  into  two  tetrapyramids,  one 

/      P        P\ 
of  which  is  at  the  right  end  of  the  axis!  +r—,—r—]  and    the 

V          2  2/ 

/        P  P\ 

other  at  its  left  end  (  +/-,  -A). 

2  2 


FIG.  194.  FIG.  195. 

Right-  (Fig.  104)  and  left- (Fig.  195)  handed  crystals  of  tartaric  acid  with  ooPoo 
)'  r^j~  (P)t^f  (P'\  oP(c),— P36  (r),  r?~  (q  in  Fig.  194)  and  /^-  (q  in  Fig.  195). 


The  prisms,  the  clino-domes,  and  the  clino-pinacoids  likewise 
may  give  rise  to  right  and  left  forms,  but  the  forms  with  planes 
that  are  parallel  to  the  &-axis  can  yield  no  new  hemimorphs. 

In  all  cases  the  corresponding  right  and  left  forms  are  enantio- 
morphous. 

On  the  crystals  of  tartaric  acid  (C4H6O6),  represented  in 
figures  194  and  195,  the  plane  p  is  a  plane  of  the  prism  at  the  right 


124  GEOMETRICAL    CRYSTALLOGRAPHY 

ooP 

end  of  the  axis,  or  is  r ,  and  p'  a  similar  plane  at  its  left  end  or 

ooP  P  io 

/ .     The  plane  q  in  figure  195  is  / and  q  in  figure   194 

P  00 

r .     The  former  is  a  left-handed  crystal  and  the  latter  a  right- 

2 

handed  crystal,  so  called  because  their  solutions  turn  the  plane 
of  polarization  of  light  to  the  left  and  the  right,  respectively. 


CHAPTER  XL 

THE  TRICLINIC  SYSTEM. 
(Pinacoidal  Class.) 

Symmetry  of  the  Triclinic  System. — The  forms  belonging 
to  the  holohedral  division  of  the  triclinic  system  possess  no  planes 
of  symmetry.  There  is,  however,  a  center  of  symmetry.  Each 
form,  therefore,  consists  simply  of  a  pair  of  parallel  faces.  Every 
pair  of  planes  on  triclinic  crystals  must  therefore  be  represented 
by  a  different  symbol. 

Axes. — As  there  are  no  planes  of  symmetry  in  this  system, 
the  axes  to  which  the  forms  are  referred  must  be  chosen  arbi- 
trarily. In  the  systems  possessing  three 
or  more  planes  of  symmetry  all  the  axes 
are  determined  by  the  symmetry.  In 
the  monoclinic  system  there  is  only  one 
plane  of  symmetry,  hence  the  position  of 
one  axis  only  is  determined.  In  the 
triclinic  system  there  are  no  planes  of 
symmetry.  Consequently,  the  position 
of  no  axis  is  fixed.  Convenience  alone 
determines  the  choice  of  axes.  FlG-  ^6 -Axes  of  the 

triclinic  system. 

Practically,  certain  planes  are  assumed 

as  prisms  or  pinacoids,  and  lines  parallel  to  these  are  made  the 
axes.  Each  substance  crystallizing  in  this  system  has  a  different 
set  of  axes,  which,  however,  when  once  established  become  the 
axes  for  all  crystals  of  that  substance. 

The  axes  in  the  system  differ,  not  merely  in  their  unit  lengths, 
but  also  in  their  inclinations.  Each  is  inclined  to  the  other  two 
at  the  point  of  their  common  intersection.  A  model  of  the  unit 
lengths  of  the  triclinic  axes  would  show  three  lines  of  different 
lengths  intersecting  each  other  at  a  common  point,  and  obliquely 
inclined  to  one  another  (Fig.  196). 

Designation  of  the  Axes. — When  the  axes  of  a  crystal  are 

I25 


126  GEOMETRICAL   CRYSTALLOGRAPHY 

once  decided  upon,  one  of  them  is  held  vertically  as  the  vertical 
axis.  Of  the  other  two,  the  longer  is  made  the  macroaxis,  b, 
and  the  shorter  the  brachyaxis,  a.  The  macroaxis  is  made  to 
incline  downward  toward  the  right  and  the  brachyaxis  down- 
ward toward  the  front.  The  signs  used  to  designate  the  axis  are 
the  same  as  in  the  orthorhombic  system.  Thus  the  symbol  of 
the  axes  in  this  system  is  a :  b  :  c. 

Groundform  and  Crystallographic  Constants. — The 
groundform  in  this  system  consists  of  two  parallel  planes  in 
opposite  octants,  each  cutting  the  axes  at  the  same  distances, 
which,  however,  are  different  on  the  different  axes.  The  inter- 

'  cepts  of  these  planes  are  the  unity  lengths  for  the  substance  on 
which  the  form  occurs.  As  in  the  other  systems  with  unequal 
axes,  the  unities  on  a  and  c  are  measured  in  terms  of  the 
unity  on  b.  This  axial  ratio  varies  for  different  substances. 

In  order  that  the  position  of  the  axes  be  fixed  it  is  necessary 
to  know  their  angles  of  inclination  to  one  another,  just  as  in  order 
to  describe  the  positions  of  the  a  and  the  c  axes  in  the  monoclinic 
system  it  is  necessary  to  specify  the  value  of  the  angle«  /?. 
In  the  triclinic  system  each  of  the  axes  is  inclined  to  the  other 
two,  hence  it  becomes  necessary  to  determine  the  values  of  three 

-  angles.  These  are  designated  as  the  angles  a,  /?,  and  r(see 
Fig.  196).  The  angle  a  is  included  between  the  vertical  and  the 

k  macroaxes.  The  angle  /?  between  the  axes  c  and  a,  and  the 
angle  7-  between  the  axes  a  and  b. 

The  crystallographic  constants  in  this  system  thus  comprise 
six  elements;  viz.:  the  lengths  of  the  three  axes,  and  the  angles 
a,  /?,  r-  For  jilbite  (NaAlSi3O8)  the  crystallographic  constants 
are  a  :  b  :  £=.6338  :  i  :  .  5577,  and  a  =94°  3',  j0=n6°  28  5/6', 
r=  88°  8  2/3'. 

Pyramids. — The  pyramids  of  the  triclinic  system  differ  but 
little  in  the  nature  of  their  planes  from  the  pyramids  in  the  mono- 
clinic  system.  Their  planes  cut  the  three  axes  at  different  dis- 
tances. If  these  distances  are  those  that  are  assumed  as  the  unity 
distances,  the  pyramid  is  a  unit  pyramid.  If  the  intercept  on  one 
of  the  lateral  axes  is  greater  than  unity  on  this  axis  when  it  is 
unity  on  the  other,  the  pyramid  is  a  br  achy  pyramid  or  a  macro- 


THE    TRICLINIC    SYSTEM  127 

pyramid  according  as  the  greater  intercept  is  on  the  macroaxis 
or  on  the  brachyaxis. 

Since  there  is  no  plane  of  symmetry  in  the  triclinic  system, 
the  existence  on  a  crystal  of  one  pyramidal  plane  necessitates  the 
presence  of  no  other  corresponding  plane  except  that  which  is 
parallel  to  it  in  the  opposite  octant,  this  being  demanded  by  the 
center  of  symmetry.  Consequently,  a  complete  geometrical  bi- 
pyramid  comprises  four  pairs  of  parallel  planes,  each  pair  of  which 
must  be  represented  by  a  distinct  symbol  (Fig.  197).  As  each 
pair  of  planes  constitutes  one-quarter  of 
a  complete  pyramid,  it  is  known  as  a 
tetrapyramid,  and  is  represented  in  the 
unit  series  by  P  with  an  accent  written 
near  (Fig.  197)  it  in  a  position  correspond- 
ing to  the  position  of  the  front  octant  in 
which  the  plane  occurs.  Thus  P'  repre- 
sents the  form,  one  of  whose  planes  is  in 
the  upper  right-hand  octant  in  front. 
P/  the  form  whose  front  plane  is  in  the 

,  .    i,   ,         ,  -r,     ,,          -  FIG.  107. — The  four  triclinic 

lower   right-hand    octant,    ,P    the    form  tetrapytamids. 

whose  front  plane  is  in  the  lower  left- 
hand  octant,  and  'P  the  form  one  of  whose  planes  is  in  the 
upper  left-hand  octant.  /P'  n  and  /P/  n  are  the  corresponding 
macrotetrapyramids  and  brachytetrapyramids.  The  indices  of 
these  forms  are  the  same  as  those  of  the  planes  on  the  front  of  the 
pyramids  of  the  tetragonal,  orthorhombic,  and  monoclinic  systems. 
Prisms. — Since  the  prisms  are  composed  of  planes  parallel 
to  the  vertical  axis,  it  necessarily  follows  that  these  planes  are  at 
the  same  time  in  two  octants.  Each  prism  consists  of  two  paral- 
lel planes  opposite  each  other.  There  are  thus  in  this  system  two 
hemiprisms  whose  symbols  are,  respectively,  QO  P,,  and  oo ,  P  f or  the 
unit  series,  oo  ~P',n  and  oo  'Pn  for  the  mac rohemi prisms  and  <x>P',n  and 
GC  'Pn  for  the  brachyhemiprisms.  The  forms  oo  P,,  GO  P, n  and  QO  P,  n 
have  planes  on  the  right-hand  side  of  the  front  of  the  crystal,  and 
QO  ,P,  oo  fjj*n  and  oo  'Pn  have  front  left-hand  planes.  The  form  oo  P,, 
and  the  corresponding  macro-  and  brachy-forms  are  derived  from 
P'  or  from  P/  by  the  change  of  the  intercept  on  c  tooo,  and  the 


128  GEOMETRICAL    CRYSTALLOGRAPHY 

forms oo 'P,  etc.,  from  ,P  or  from  'P  in  the  same  manner.     The 
corresponding  indices  of  the  hemiprisms  are: 

oo  P',  iio;ocP'w,  Mo;ocP,n,  kho 

oc'P,  iio;<x,'Pn,  Mo;oo!Pn,  kho 

Domes. — The  macrodome  planes  that  occur  on  the  front  of 
triclinic  crystals  lie  either  in  the  two  upper  octants  or  in  the  two 
lower  ones.  They  may  thus  be  regarded  as  derived  from  T  and 
P'  or  from  /P  and  P/.  Each  consists  of  a  pair  of  faces,  hence  each 
corresponds  to  a  half  of  an  orthorhombic  macrodome.  Their 
symbols  are  'P'ocfor  the  upper  front  hemimacrodome  and, 
,P,oo  for  the  lower  front  hemimacrodome  (Fig.  198).  Their 
corresponding  indices  are:  hoi  and  hoi. 


FIG.  198. — Two  triclinic  FIG.  199. — Two  triclinic 

macrodomes.  brachydomes. 

Of  the  brachydomes  there  are  also  two,  each  consisting  of  a 
single  pair  of  planes.  Since  their  planes  are  parallel  to  the  a  axis, 
each  is  at  the  same  time  in  two  octants.  The  lower  planes  are 
parallel  to  the  opposite  upper  ones.  One  of  the  planes  is  in  the 
upper  right-hand  octant  in  front  and  in  the  corresponding  octant 
behind.  But  the  octant  behind  corresponds  to  the  lower  left- 
hand  octant  in  front.  Consequently,  this  dome  face  is  regarded 
as  derived  from  the  pyramidal  faces  corresponding  to  /P',  hence  its 
symbol  is  ,P'  oo  or  ohl.  This  is  the  upper  right-hand  hemibrachy- 
dome  or  the  lower  left-hand  one.  Similarly  T/  oo  is  the  upper 
left-hand  brachydome  or  the  lower  right-hand  one  (Fig.  199),  with 
the  indices  ohl. 

Pinacoids. — The  pinacoidal  planes  are  parallel  to  two  axes, 
consequently  each  lies  in  four  octants.  There  is  only  one  brachy- 


THE    TRICLINIC    SYSTEM 


129 


pinacoid,  only  one  macropinacoid,  and  only  one  basal  pinacoid 
possible  on  a  triclinic  crystal.  Their  symbols  are,  respectively, 
ooP  oo,  ocPoo,  and  oP,  or  oio,  100,  and  ooi. 

Series  of  Pyramids  and  Domes. — In  the  discussion  above  it 
is  assumed  that  the  pyramidal  and  dome  faces  cut  the  vertical 
axis  at  unity.  In  this  system,  however,  as  in  the  monoclinic  and 
orthorhombic  systems,  there  occur  prisms  and  domes  whose  inter- 
cepts on  the  c  axis  are  some  multiple  of  unity.  Their  symbols 
may  be  represented  by  prefixing  m  to  the  symbols  of  the  corre- 
sponding unit  forms,  or  by  indices  that  are  the  same  as  those 
which  represent  the  front  planes  on  corresponding  forms  in  the 
orthorhombic  system. 

Combinations  of  Forms. — Every  pair  of  faces  in  triclinic 
crystals  demands  a  separate  symbol  for  its  representation,  hence 


FIG.  200. — Crystal  of  cop- 
per sulphate.  See  text  for 
symbols  of  planes. 


FIG.  201.  —  Crystal  of 
anorthite.  See  text  for  sym- 
bols of  planes. 


the  symbols  of  these  crystals  are  usually  long  and  somewhat  com- 
plicated. The  difficulty  of  deciphering  triclinic  crystals  is  in- 
creased by  the  fact  that  there  is  almost  an  unlimited  choice  in  the 
selection  of  the  axes,  and  that  frequently  the  choice  made  for 
many  crystals  is  not  the  most  convenient.  A  few  illustrations 
may  indicate  the  principles  made  use  of  in  fixing  the  axes.  Fig- 
ure 200  is  a  crystal  of  blue  vitriol  (CuSO4  +7H2O).  Its  planes 
are  oP(c);«P<»(^);ooP»(6);oopi  (n);ao',P(m);<x>',P2(l)  and  P'(s). 
Figure  201  represents  a  crystal  of  anorthite  (CaAl2Si2O8).  It  is 
much  more  complicated  than  the  blue  vitriol  crystal  as  it  contains 
the  forms  oP(P);  ooP^(M);  ooP^/);  ooIPfT);  ooP^f/);  00^3(2); 
9 


130  GEOMETRICAL   CRYSTALLOGRAPHY 

F(iii);  'P(a);  P,  (*);  ,P  (#) ;  •  4P<  2  (v) ;  T'^(0;  2,P,«  (y); 
,P'  *  («) ;  2,P'  5  (r)  and  T, » (»). 

Comparison  of  the  Systems. — The  main  differences  between 
the  crystallographic  systems  are  conditioned  by  differences  in 
the  symmetry  of  the  holohedrons  belonging  to  them.  The 
differences  are  expressed  primarily  in  the  character  of  the  axes 
chosen  for  each  system.  For  that  system  possessing  the  highest 
grade  of  symmetry  the  unities  on  all  the  axes  are  of  equal  length 
and  the  axes  themselves  are  fixed  in  position,  being  perpendicular 
to  one  another.  In  the  system  possessing  the  next  lower  grade 
of  symmetry  there  are  four  axes,  one  of  which  is  of  different 
length  from  the  other  three.  All  are  fixed  in  position.  The 
three  axes  of  the  same  length  are  inclined  to  each  other  at  angles 
of  60°  and  are  perpendicular  to  the  fourth  axis.  The  system  of 
the  next  lower  grade  has  three  axes,  one  of  which  is  different  in 
length  from  the  other  two  which  are  equal.  All  are  at  right 
angles  to  one  another.  These  three  systems  are  all  characterized 
by  the  possession  of  at  least  one  principal  plane  of  symmetry. 

Of  the  systems  having  no  principal  plane  of  symmetry,  that  of 
the  highest  grade  of  symmetry  possesses  three  axes  of  different 
lengths  perpendicular  to  one  another.  The  next  system,  which 
has  only  one  symmetry  plane,  has  only  one  of  its  axes  fixed  in 
position.  This  is  perpendicular  to  the  plane  of  symmetry.  The 
other  two  axes  are  in  the  symmetry  plane,  but  their  positions  in 
this  plane  are  not  fixed.  Lines  lying  within  this  plane  are 
arbitrarily  chosen  to  serve  as  axes.  The  axes,  therefore,  are 
three  unequal  lines;  one  at  right  angles  to  the  plane  of  the 
other  two.  The  system  with  the  lowest  grade  of  symmetry  has 
no  axes  fixed  for  it.  They  are  chosen  arbitrarily  and  must 
consist  of  three  unequal  lines  inclined  to  one  another  at  angles 
other  than  right  angles. 

The  symbols  for  the  axes  become  more  and  more  complex 
as  we  pass  from  the  systems  of  higher  to  lower  grade,  and  more 
variable  factors  are  concerned  in  them.  These  facts  may  be 
expressed  by  writing  the  symbols  as  follows,  omitting  angles 
of  90°: 


THE    TRICLINIC    SYSTEM  131 

Systems.  Axes 

Isometric a 

Hexagonal a  :  c 

Tetragonal a  :  c 

Orthorhombic a  :  b  :  c 

Monoclinic ; .  a  :  b  :  c ;  /? 

Triclinic a  :  b  :  c ;  a,  ft  7-. 


CHAPTER  XII. 
CRYSTAL  IMPERFECTIONS. 

Ideal  Forms. — Occasionally  a  crystal  occurs  which 
possesses  the  regularity  of  a  well-made  model.  All  the  faces 
belonging  to  the  same  form  have  the  same  size.  They  are  all 
equally  developed  and  are  all  at  equal  distances  from  the  center. 
These  are  ideal  forms  like  those  represented  by  most  of  the 
figures  in  this  book.  Usually,  however,  the  regular  and  sym- 
metrical growth  of  a  crystal  has  been  interfered  with  by  some 
external  agency  or  condition  unfavorable  to  the  symmetrical 
development  of  the  ideal  form.  Moreover,  it  frequently  happens 
that  the  faces  of  crystals  are  not  perfectly  plane  as  they  are 


FIG.  202. 


FIG.  203. 


FIG.  204. 


Symmetrical  and  distorted  cubes  with  columnar  (Fig.  203)  and  tabular 
(Fig.  204)   habits. 

assumed  to  be  in  crystallographic  discussions,  and  occasionally 
the  values  of  the  interfacial  angles  vary  from  the  calculated  values 
for  the  ideal  form.  These  are  all  imperfections  in  the  crystals. 
They  may  be  classified  as  imperfections  in  symmetrical  develop- 
ment, imperfections  in  faces,  and  imperfections  in  angles. 

Distorted  Crystals. — The  unequal  rapidity  of  crystal  growth 
in  different  directions  often  results  in  the  elongation  of  the  crystal 
in  certain  directions,  and  the  consequent  increase  in  the  size  of 
some  of  its  planes  at  the  expense  of  others.  This  process  when 
carried  to  excess  actually  crowds  out  some  of  the  faces  that  should 

132 


CRYSTAL   IMPERFECTIONS  133 

appear  on  the  crystal,  apparently  destroys  its  symmetry,  and 
produces  bodies  with  the  shapes  characteristic  of  crystals  of  a 
lower  grade  of  symmetry.  The  cube  may  grow  much  faster  in 
the  direction  of  one  axis  than  in  the  direction  of  the  others,  and 
thus  may  simulate  a  tetragonal  prism  and  basal  plane,  or  it  may 
grow  rapidly  in  two  directions,  producing  a  tabular  crystal  (Figs. 
202,  203,  and  204).  The  dodecahedron  of  the  isometric  system 


FIG.  205.  FIG.  206.  FIG.  207. 

Dodecahedron  distorted  in  such  a  way  as  to  appear  to  possess  tetragonal 
(Fig.  205),  hexagonal  (Fig.  206),  and  orthorhombic  (Fig.  207)  symmetry. 

may  by  elongation  in  different  directions  come  to  resemble  a 
tetragonal  crystal  (Fig.  205),  an  hexagonal  one  (Fig.  206),  or  an 
orthorhombic  one  (Fig.  207).  By  elongation  in  two  directions, 
that  is,  by  flattening,  tabular  crystals  may  be  produced  which 
at  the  first  glance  do  not  resemble  in  the  slightest  degree  the 
forms  from  which  they  were  derived.  Figure  208,  for  instance, 
illustrates  an  octahedron  that  has  been  flattened  in  the  direction 
of  a  line  passing  through  the  centers  of  opposite  octahedral  faces. 

Nevertheless,  however  much  crystals  may 
be  distorted  in  this  way,  their  interfacial  angles 
always  have  the  values  possessed  by  the  cor- 
responding   ideal    forms    and    their    physical 
properties   are  the  same.      Moreover,  by  im- 
agining the  corresponding  faces  on  all  sides     FIG.  208.— Flattened 
of  the  crystal  to  be  moved  toward  its  center, 
the  form  may  be  restored  to  the  shape  and  symmetry  properly 
belonging  to  it. 

Habit  of  Crystals. — The  habit  of  a  crystal,  or  its  general 
shape,  depends  largely  upon  the  character  of  the  distortion  it  has 
suffered.  It  is  often  so  characteristic  a  feature  of  different  sub- 
stances that  a  number  of  terms  have  been  invented  for  trie 
purpose  of  describing  it. 


134 


GEOMETRICAL   CRYSTALLOGRAPHY 


When  a  crystal  grows  much  more  rapidly  in  one  direction 
than  in  others,  and  consequently  is  elongated  in  a  single  direction, 
it  is  said  to  have  an  acicular  habit  or  to  be  acicular,  when  its  cross 
section  is  very  small;  to  be  prismatic  or  columnar,  when  its  cross 
section  is  greater.  Quartz  crystals  are  columnar  in  the  direction 
of  the  c  axis,  orthoclase  in  the  direction  of  the  a  axis,  and  epidote 
in  the  direction  of  the  b  axis.  Figures  209,  210,  and  211  represent 
crystals  of  these  three  substances. 

When  the  elongation  is  in  the  direction  of  two  crystallographic 
axes  or,  more  properly,  when  the  crystal  is  most  largely  developed 
laterally,  it  is  tabular,  as  are  many  crystals  of  mica. 


oP 


031*00 


FIG.  209. 


FIG.  210. 


FIG.  211. 


FIG.  209. — Columnar  crystal  of  quartz  elongated  parallel  to  c. 
FIG.  210. — Columnar  crystal  of  feldspar  elongated  parallel  to  a. 
FIG.  211. — Columnar  crystal  of  epidote  elongated  parallel  to  &. 


Other  terms  besides  those  defined  are  used  in  the  descriptions 
of  the  habits  of  crystals,  but  they  are  self-explanatory  and  so  need 
no  special  definition. 

Deformed  Crystals. — Distortions  of  another  kind  often 
produce  very  peculiar  results,  such  as  twisted  or  curved  crystals, 
bent  crystals,  rounded  crystals,  etc.,  all  of  which  are  caused 
either  by  the  prevalence  of  very  unusual  conditions  during  growth 
or  by  the  action  upon  the  crystal  of  mechanical  forces  after  its 
formation.  Many  crystals  imbedded  in  rocks  are  bent,  and 
often  they  are  cracked  or  shattered.  Distortions  often  affect 
not  only  the  shapes  of  the  crystals,  but  often  their  interfacial 
angles  as  well.  Figure  212  is  a  picture  of  a  curved  hornblende 


CRYSTAL   IMPERFECTIONS 


crystal  in  a  rock  from  near  Marquette  in  Michigan.     The  rock  is 
magnified  30  diameters. 

Imperfections  in  Crystal  Planes. — Although  theoretically 
crystal  faces  are  plane  surfaces,  practically  a  very  large  proportion 


FIG.   212.— Curved  crystal  of  hornblende  in  rock.     (After  G.  H.  Williams.} 

of  the  crystal  planes  found  in  nature  are  not  even.  The  unevenness 
may  be  due  to  striations,  curvature,  corrosion,  or  irregularity  in 
growth. 

Striations. — Striations  are  fine  parallel  lines  that  sometimes 
extend  across  all  the  faces  of  a  crystal  and  at  other  times  are 


FIG.  213. — Striated 
quartz  crystal. 


FIG.  214. — Striated 
crystal  of  pyrite. 


FIG.   215.— Striated 
crystal  of  plagioclase. 


limited  to  the  planes  belonging  to  a  single  form.  These  may 
originate  in  the  oscillatory  combination  of  two  distinct  forms, 
as  the  horizontal  striations  on  the  prismatic  faces  of  many 
quartz  crystals  (Fig.  213),  which  are  produced  by  the  oscillatory 


i36 


GEOMETRICAL    CRYSTALLOGRAPHY 


combination  of  ooR  and  R,  or  on  cubes  of  pyrite  (Fig.  214)  due 

[wOool 
to  combinations  of  oo  O  oo  and  -   .  They  may  also  be  due  to 

polysynthetic  twinning,  as  in  the  case  of  striations  on  the  prismatic, 
the  basal,  and  the  macrodome  faces  of  plagioclase  crystals 
(Fig.  215),  when  the  twinning  lamellae  are  so  small  that  they  can 
with  difficulty  be  recognized.  (See  p.  155-156.) 

Curvature. — Many  substances,  like  diamond  (C)  (Fig.  216), 
dolomite  ((CaMg)CO3),  etc.,  are  so  frequently  bounded  by 
curved  faces  that  their  occurrence  must  be  ascribed  to  some 
characteristic  property  of  the  molecules  of  which  these  bodies 
are  composed. 


FIG.   216. — Crystal  of  diamond 
with  curved  faces. 


FIG.   217. — Natural    etched  figures  on 
diamond  crystal.     (After  Tschermak.) 


Corrosion  and  Etched  Figures. — After  a  crystal  has  been 
formed  with  perfect  planes,  solutions  may  attack  it  and  dissolve 
portions  of  the  faces,  rounding  their  edges  and  so  causing  the 
plane  to  become  curved,  or  else  pitting  them  with  little  hollows, 
known  as  etched  figures.  The  etched  figures  vary  in  shape  in 
different  minerals  and  on  different  planes  of  the  same  mineral, 
their  shapes  always  being  governed  by  the  symmetry  of  the  plane 
on  which  they  occur.  Those  on  the  cubic  faces  of  diamond 
(Fig.  217)  are  hopper-shaped. 

Imperfections  Caused  by  Irregularities  in  Growth.— 
The  irregularities  developed  during  the  growth  of  a  crystal  are 
too  numerous  to  be  specified.  Often  an  obstruction  met  with  by 
the  crystal  will  become  imbedded  in  it  and  so  cause  one  or  more 
of  its  planes  to  become  distorted,  or  it  may  impress  its  shape  upon 


CRYSTAL   IMPERFECTIONS 


137 


a  plane  with  which  it  comes  in  contact  and  in  this  way  destroy  its 
perfection  of  surface.  Or  a  crystal  may  grow  rapidly,  forming  its 
edges  first  and  building  up  a  skeleton,  which  may  not  become 
entirely  filled  up.  In  place  of  a  crystal  face  there  may  thus  result 
a  reversed  pyramid,  as  in  the  so-called  hopper-shaped  crystals  of 
salt,  crystals  of  cuprite  (CuO)  (Fig.  218),  or  galena  (PbS),  and  in 
metallic  bismuth  (Fig.  219). 

Symmetry  of  Imperfections. — Although  the  imperfections 
in  crystal  faces  tend  to  destroy  their  ideal  development,  these 
imperfections  are  nevertheless  governed  in  a  great  measure  by  the 
symmetry  of  the  crystal  form.  All  cry stallo graphically  equivalent 


FIG.   218. — Skeleton  crystal 
of  cuprite. 


FIG.   219. — Portion  of  skeleton 
crystal  (cube)  of  bismuth. 


planes  are  similarly  affected.  The  symmetry  of  the  imperfections 
on  forms  in  combinations  or  even  in  apparently  simple  forms  may 
often  serve  to  aid  in  the  correct  determination  of  their  sym- 
metry. In  this  way  certain  forms  that  appear  to  be  holphedral 
are  shown  to  be  in  reality  hemihedrons  that  are  not  geometric- 
ally distinguishable  from  the  holohedrons  from  which  they  are 
derived. 

The  striations  on  cubes  of  pyrite  (FeS2)  (see  Fig.  214)  are 
often  arranged  symmetrically  with  respect  to  the  three  principal 
planes  of  symmetry  passing  through  the  cube,  and  with  respect  to 
these  only.  The  cube  is  thus  symmetrical  in  the  same  way  as  are 
the  hemihedrons  derived  by  parallel  hemihe-drism,  hence  the  form 
may  be  regarded  as  a  hemihedral  cube  derived  from  the  holo- 


138 


GEOMETRICAL    CRYSTALLOGRAPHY 


FIG.  220. — Crystal 
showing  distribution 
factions. 


Sp) 

of    galena, 
of     imper- 


hedral  cube  by  the  parallel  method  of  hemihedrism.  When  in 
combination  with  distinctive  hemihedrons,  these  are  always 
either  the  diploid  or  the  pyritoid,  the  characteristic  hemihedrons 
of  parallel  hemihedrism,  and  never  forms  derived  by  the  gyroidal 
or  the  inclined  methods  of  hemihedrism. 

Again,  on  the  mineral  sphalerite  (ZnS)  there  frequently  occur 
eight  triangular  faces  distributed  like  the  eight  faces  of  an  octa- 
hedron (Fig.  220).  But  of  these,  four  are  smooth  except  for  a  few 

striations,  and  four  are  rough,  a 
smooth  and  a  rough  face  alternat- 
ing. The  rough  faces  correspond 
to  the  faces  of  a  tetrahedron  and  the 
smooth  ones  to  those  of  a  corre- 
sponding tetrahedron  with  an  op- 
posite sign.  If  the  former  be  con- 
sidered +  ,  the  latter  is  —  . 

2  2 

These  forms  are  inclined  hemihe- 
drons, hence  we  should  expect  all 
other  forms  on  this  same  mineral  to  be  inclined  hemihedrons 
with  characteristic  forms,  or  holohedrons  that  do  not  yield  new 
forms  by  the  inclined  method  of  hemihedrism.  As  a  matter  of 
fact,  these  are  the  only  forms  that  do  occur  in  the  mineral. 

Variations  in  Crystal  Angles. — The  imperfections  in  the 
planes  on  crystals  do  not  often  affect  the  values  of  their  interfacial 
angles.  When  these  vary  the  variations  are  usually  due  to  a 
difference  in  the  chemical  composition  of  the  minerals  whose 
angles  are  being  compared,  to  differences  in  the  temperature  at 
which  the  crystals  are  measured,  or  to  some  mechanical  force 
acting  upon  the  crystal  from  without.  The  variations  are  always 
slight  and,  except  in  the  case  where  mechanical  force  is  concerned, 
the  variations  only  emphasize  the  fact  that  the  interfacial  angles 
on  crystals  are  their  most  characteristic  features. 

Impurities  within  Crystals. — Ideal  crystals  consist  of  homo- 
geneous matter  possessing  the  same  chemical  composition  through- 
out and  having  the  color  of  the  pure  substance  that  composes  their 
mass.  Departures  from  this  ideal  purity  are  frequently  met  with. 


CRYSTAL   IMPERFECTIONS  139 

The  impurities  may  consist  of  a  dilute  coloring  matter,  or 
pigment,  which  is  present  in  such  small  quantity  as  to  be  inde- 
terminable, but  nevertheless  in  sufficient  quantity  to  give  a  de- 
cided color  to  the  crystals.  Pure  quartz  (SiO2)  is  colorless,  smoky 
quartz  is  gray  or  black,  but  the  coloring  matter  in  it  is  present  in 
such  small  quantity  that  its  true  nature  is  not  certainly  known. 
Sometimes  the  coloring  matter  is  evenly  distributed  throughout 
the  entire  crystal,  sometimes  it  is  regularly  distributed  in  concen- 
tric layers,  called  zones,  and  sometimes  it 
is  irregularly  distributed  without  respect  to 
crystallographic  directions.  Tourmaline 
crystals  from  Hebron  and  Mount  Mica, 
Maine,  very  often  have  a  pink  nucleus  sur- 
rounded by  zones  of  different  shades  of 
green.  Figure  221  represents  a  cross  sec- 
tion  through  such  a  crystal.  F[G  22I  _Cross  sec. 

The  other  impurities  in  crystals   may    tion  of  crystal  of  tour- 

,         ,  ,  ,  .      ,       .  „,.  maline    exhibiting    zonal 

be  classed  together  as  inclusions.     They  can    arrangement  of  color, 
usually  be  distinguished  from  the  mineral 
substance  in  which  they  are  imbedded,  if  not  by  the  eye,  at  least 
under  the  microscope.     They  comprise  gas,  fluid,  glass,  crystal, 
and  unindividualized  inclusions. 

Gas  inclusions  are  found  in  crystals  of  both  aqueous  and 
igneous  origin.  They  appear  as  little  cavities  of  different  shapes, 
in  some  cases  scattered  promiscuously  through  the  crystal  sub- 
stance; in  others  arranged  zonally.  When  the  cavities  have 
the  shapes  of  the  crystal  they  are  known  as  negative  crystals. 
Their  contents  may  be  air,  carbon  dioxide  (CO2),  marsh  gas 
(CH4),  or  some  other  hydrocarbon,  sulphur  dioxide  (SO2),  or 
some  simple  gas. 

Fluid  Inclusions. — When  the  pores  are  filled  with  liquid  they  are 
known  as  fluid  cavities.  Like  the  gas  inclusions  these  may  be  of 
different  shapes,  and,  like  them,  they  may  be  arranged  regularly  or 
irregularly  within  the  crystal  substance.  The  fluid  enclosed  in 
the  cavities  may  be  water,  liquid  carbon  dioxide,  some  other 
liquid  gas,  petroleum,  or  some  dilute  salt  solution.  Often  the 
liquid  encloses  a  bubble  of  gas  (Fig.  222)  which  by  its  movements 


140 


GEOMETRICAL    CRYSTALLOGRAPHY 


serves  to  distinguish  the  fluid  from  the  glass  inclusion.     Figure 
223  is  a  magnified  view  of  gas  and  liquid  inclusions  in  quartz. 

In  a  few  cases  two  fluids  which  do  not  mix  are  enclosed  in  the 
same  cavity  in  which  case  that  one  whose  surface  tension  is  greatest 
occupies  the  center  of  the  cavity,  while  the  other  surrounds  it  as 
an  envelope  (Fig.  224).  When  liquid  and  gas  are  both  enclosed  in 
the  same  cavity,  the  size  of  the  gas  bubble  may  be  made  to 
change  by  increasing  or  diminishing  the  temperature,  and  at  the 
proper  temperature  may  be  made  to  disappear  entirely.  By 
noting  the  temperature  at  which  the  bubble  disappears,  i.e.,  the 


FIG.   222. — Liquid  inclusions  in 
quartz.     Slightly  magnified. 


FIG.   223.- — Liquid  inclusions  in 
quartz.     Greatly  magnified. 


temperature  at  which  there  is  no  distinction  between  the  liquid 
and  the  gas,  the  nature  of  the  inclusion  may  sometimes  be 
determined. 

These  phenomena  are  supposed  to  show  that  the  substance 
was  crystallized  under  high  pressure,  and  that  while  growing  it 
surrounded  a  portion  of  the  mother  liquor  from  which  it  was 
separating. 

Glass  inclusions  occur  in  crystals  that  have  solidified  from  a 
molten  mass  and  have  enclosed  a  portion  of  this  mass  during 
growth,  or  in  those  which  have  since  their  formation  been  heated 
so  high  as  to  melt  foreign  substances  contained  within  them. 
These  inclusions  also  often  contain  bubbles,  but  in  this  case  they 
are  immovable.  The  existence  of  two  bubbles  in  the  same 
inclusion  is  proof  that  the  cavity  is  filled  with  glass. 


CRYSTAL   IMPERFECTIONS  141 

Crystal  inclusions. — These  are  minute  crystals  caught  up  and 
enclosed  in  larger  crystals  during  growth,  or  minute  crystals 
formed  by  the  decomposition  of  the  latter. 


FIG.  224.— Thin  section  of  smoky  quartz  showing  inclusions  consisting  of  central 
gas  bubble,  surrounded  by  two  liquids  that  do  not  mix.  The  outer  liquid  is 
probably  water  and  inner  liquid  and  gas  bubble  carbon  dioxide.  Magnified  about 
150  diam.  (After  Rosenbusch.) 

When  a  crystal  of  orthoclase  solidifies  from  a  solution  in  which 
there  occur  already  formed  tiny  crystals  of  apatite,  some  of  these 
small  crystals  may  become  embedded  in  the  orthoclase  as  inclu- 
sions. On  the  other  hand,  when  orthoclase  under  the  influence 


FIG.  225. — Section  of  leucite  crystal  showing  zonal  arrangement  of  minute 
inclusions.     (After  Tschermak.} 

of  chemical  agents  begins  to  change  into  kaolin,  the  first  stages  of 
the  alteration  are  seen  in  the  presence  of  tiny  scales  of  kaolin 
scattered  through  the  mass  of  the  orthoclase.  In  some  crystals 
the  inclusions  are  so  arranged  as  to  produce  an  iridescence  in  the 


142  GEOMETRICAL    CRYSTALLOGRAPHY 

enclosing  substance,  as  in  the  case  of  labradorite,  bronzite,  etc. 
In  others  they  are  arranged  zonally,  as  in  leucite  (Fig.  225) .  When 
so  small  as  to  baffle  attempts  to  identify  them  as  definite  mineral 
species,  the  inclusions  are  known  as  crystallites;  when  large  enough 
to  be  identified,  they  are  referred  to  as  microlites. 

The  inclusions  in  crystals  can  best  be  studied  in  very  thin 
slices.  When  these  are  viewed  under  the  microscope  it  is  found 
that  very  few  crystals  are  free  from  impurities  of  many  kinds. 
Nearly  all  contain  either  liquid,  gas,  or  mineral  inclusions.  The 
former  are  especially  abundant  even  in  crystals  occurring  in  the 
hardest  rocks.  Often  in  a  single  grain  (Fig.  222)  of  the  quartz  in 
a  granite  will  be  seen  hundreds  of  tiny  pores  nearly  filled  with  liquid. 
In  many  of  these  are  little  bubbles  that  move  slowly  through  the 
liquid  mass  or  dance  rapidly,  as  the  case  may  be,  moving  inces- 
santly, in  consequence,  probably,  of  the  slight  changes  of  tempera- 
ture to  which  the  substance  containing  them  is  subjected.  Some- 
.times,  but  not  as  commonly,  a  liquid  inclusion  will  contain  a 
little  crystal  of  some  salt,  which  may  be  made  to  dissolve  by 
warming  the  specimen  or  to  grow  larger  by  cooling  it. 

The  quantity  of  liquid  enclosed  in  some  minerals  is  very 
large,  reaching,  it  is  said,  in  the  case  of  the  constituents  of  certain 
rocks  as  much  as  i .  8  per  cent,  of  their  volume.  Often  the  liquid 
is  really  a  condensed  gas,  most  commonly  liquid  carbon  dioxide, 
methane,  or  nitrogen.  The  volume  of  gases  obtained  from  en- 
closures in  quartz  crystals  from  Poretta,  Italy,  amounted  in  one 
instance  to  .  03  per  cent,  of  the  volume  of  the  quartz.  The  total 
quantity  of  gas  that  has  been  obtained  from  certain  other  crystals 
is  much  greater,  reaching  13  per  cent,  by  volume  in  some  cases, 
but  much  of  this  was  present  in  some  other  form  than  as  enclosures 
in  cavities. 


CHAPTER  XIII. 
CRYSTAL  AGGREGATES. 

Crystal  Individuals  and  Crystal  Aggregates.— Thus  far 

only  crystal  individuals  have  been  discussed,  since  these  most 
nearly  exhibit  the  ideal  forms  demanded  by  symmetry.  Rarely, 
however,  are  crystal  individuals  complete.  There  is  usually 
lacking  some  one  or  more  of  their  faces  where  the  crystal  was 
attached  during  its  growth. 

Individual  crystals  that  exhibit  only  an  occasional  face  and 
crystallized  masses  that  possess  forms  due  to  influences  acting 
from  without  the  crystallizing  body  (allotriomorphic  forms)  are 
distinguished  as  crystal  grains. 

Neither    crystal    individuals    nor    crystal    grains    commonly 
occur  isolated.     They  are  usually  grouped  together  irregularly. 
Often,  however,  several  crystals  or  parts  of  crystals  are  grouped 
in  parallel  positions,  or  in  such  a  way  that  the  different  individuals 
are  separated  from  one  another  by  a  plane  about  which  they  are ' 
symmetrically  disposed.     The  former  of  these  definite  groupings 
are  known  as  parallel  growths  and  the  latter  as  twinned  crystals.  l 
The  general  term  "aggregate"  is  given  to  the  irregular  groupings.  „ 

Irregular  Aggregates. — Irregular  aggregates  may  consist  of 
grains  of  the  same  mineral  irregularly  grouped,  as  are  the  particles 
of  calcite  in  a  coarse-grained  limestone  (Fig.  226),  or  they  may  be 
composed  of  grains  of  several  substances,  as  is  the  aggregate  of 
orthoclase  (KAlSi3O8),  quartz  (SiO2),  and  mica  known  as 
granite  (Fig.  227).  Such  aggregates  as  these,  where  each  grain 
is  completely  or  almost  completely  bounded  by  other  grains,  are 
distinguished  as  crystalline  aggregates. 

When  the  grains  in  an  irregular  aggregate  are  partially 
bounded  by  their  own  planes,  the  aggregate  is  a  crystal  aggregate; 
and  when  the  individual  grains  approach  in  form  the  character  of 
crystal  individuals,  the  aggregate  is  a  crystal  group. 


144 


GEOMETRICAL    CRYSTALLOGRAPHY 


Figure  227  is  from  an  enlarged  photograph  of  a  thin  slice 
of  granite.     It  shows  the  minerals  crowded  together  irregularly. 


FIG.  226. — Piece  of  white  marble,  a  crystalline  aggregate  of  a  single  mineral 
(calcite).  The  bright  areas  are  due  to  reflections  from  the  cleavage  surfaces  of 
individual  grains. 


FIG.  227. — Thin  section  of  granite,  illustrating  a  granular  aggregate  of  several 
different  minerals,  viz:  quartz  (clear),  soda-feldspar  (striped),  potash  feldspar 
(cross-barred)  and  biotite  (black).  Magnified  about  5  diameters. 

None  of  them    possess   crystal   forms.     Figure    i    (page   3)    is 
a   crystal    group,  composed  of   well-defined  crystal   individuals 


CRYSTAL  AGGREGATES  145 

of  calcite,  each  bounded  by  its  own  planes  except  at  its  point 
of  attachment. 

Classification  of  Crystalline  Aggregates. — The  classifica- 
tion of  crystalline  aggregates  is  not  in  itself  a  matter  of  great 
importance.  In  discussing  aggregates  of  crystal  particles, 
however,  it  is  convenient  to  make  use  of  terms  that  will  indicate 
briefly  what  would  in  ordinary  language  require  a  clause  for  its 
description.  For  descriptive  purposes,  then,  we  may  classify 
aggregates  with  respect  to  the  size  of  their  constituent  grains, 
with  respect  to  their  manner  of  development,  and  with  respect  to 
the  strength  of  their  cohesion. 


FIG.  228. — Radiating  groups  of  wavellite  crystals  on  a  rock  surface. 

According  to  the  size  of  the  component  grains,  an  aggregate 
may  be  phanero-crystalline  when  its  particles  are  large  enough  to 
be  seen  by  the  naked  eye,  or  crypto-crystalline  when  they  may  be 
detected  only  with  the  aid  of  a  microscope.  Phanero-crystalline 
aggregates  may  be  coarse-grained,  medium-grained,  or  fine- 
grained. 

According  to  the  manner  of  development  of  its  individual 
components,  an  aggregate  may  have  a  structure,  which  is  described 
as  granular,  when  the  particles  are  about  equally  developed  in  all 
directions,  as  are  the  grains  in  a  granite  or  a  coarse  limestone; 
lamellar,  or  platy,  when  the  grains  are  scaly,  platy,  or  tabular, 


146 


GEOMETRICAL    CRYSTALLOGRAPHY 


as  in  the  case  of  a  mica-schist;  or  fibrous,  when  the  components  are 
all  acicular,  as  in  the  case  of  asbestus  ((MgCa)SiO3). 

The  components  of  a  fibrous  aggregate  often  tend  to  group 


FIG.  229. — Globular  mass  of  limonite  fibers. 

themselves  around  certain  centers  from  which  they  radiate, 
producing  masses  that  imitate  the  shapes  of  common  objects. 
When  the  fibers  radiate  from  a  point  and  are  confined  to  a  single 


FIG.  230. — Botryoidal  groups  of  limonite  fibers. 

plane,  like  the  spokes  in  a  wagon  wheel,  they  produce  radial 
aggregates  (Fig.  228).  When  the  radiation  is  outward  in  all 
directions  from  a  point,  globular  forms  result  (Fig.  229).  These 


CRYSTAL  AGGREGATES 


latter  in  turn  may  aggregate,  forming  a  bunch  of  globules,  when 
its  form  is  known  as  botryoidal  (Fig.  230).  Often  the  radial 
grouping  is  incomplete  or  the  fibers  may  not  diverge  regularly. 
In  this  case  a  sheaf-like  bundle  may  result,  as  in  the  case  of  the 


FIG.  231.— Sheaf-like  group  of  stilbite  crystals. 

mineral   stilbite    (calcium   aluminium  silicate  with  water)   (Fig. 

231)- 

When  the  fibers  radiate  from  a  line  they  produce  a  stalactitic 
growth,  or  a  stalactite  (Figs.  232  and  233). 


FIG.  232. — Stalactite  of  limonite. 

If  the  cohesion  between  the  particles  of  an  aggregate  is  strong, 
whatever  may  be  the  shape  of  the  aggregate,  it  is  said  to  be 
compact;  when  the  cohesion  is  slight,  the  aggregate  is  friable. 

Parallel  Growths. — Frequently  crystals  are  so  grouped  that 


148 


GEOMETRICAL   CRYSTALLOGRAPHY 


one  or  all  of  the  axes  of  the  different  individuals  are  parallel. 
The  most  complete  parallelism  occurs  in  the  case  of  individuals  of 
the  same  substance.  Often  two  crystals  of  the  same  substance 
occur  side  by  side  in  contact,  with  all  the  axes  of  the  one  parallel 


FIG.  233.  —  Cross  section  of 
group  of  stalactites  of  limonite 
and  goethite  (hydroxides  of 
iron.) 


FIG.  234. — Group  of  barite  crystals 
with  axes  parallel.    (After  Tschermak.} 


FiG.  235. —  Group  of  quartz 
crystals  merging  into  single  crystal 
below.  (After  Tschermak.] 


FIG.  236. — A  number  of  small 
crystals  of  quartz  grouped  to 
form  a  single  large  crystal. 


to  the  corresponding  axes  of  the  other,  as  in  the  case  of  barite 
(BaSO4)  (Fig.  234).  Sometimes  the  contact  surfaces  are  small, 
when  the  crystals  just  touch.  At  other  times  the  crystals  appear 
to  penetrate  one  another,  only  a  portion  of  each  crystal  being 


CRYSTAL  AGGREGATES 


149 


visible.    Figure  235  is  a  group  of  quartz  crystals  the  lower  portions 

of  which  are  merged  into  a  single  crystal.     In  some  cases  many 

small   individuals    (sub-individuals)    are   grouped   side  by  side, 

forming  a  large  crystal  (Fig.  236),  or  are  arranged  along  a  common 

axis  in  such  a  way  that  only  the  individual  at  the 

two   ends  of  the  group   exhibit  complete  faces 

(zircon,  ZrSiO4,  Fig.  237),  while  the  intermediate 

individuals    appear    as    thin    plates    or  lamellae 

crowded  between  these.     When  the  lamellae  of 

the  intermediate   individual  are  very  thin   they 

produce  the  effect  of  striations  on  the  faces  of  a 

single  individual.     These  striations  are  caused 

by    the    repetition    of    the    interfacial   edges  on 

consecutive  individuals — a  condition   known   as 

an  oscillatory  combination. 

Again  many  minerals  form  branching  aggre- 
gates, the  branches  of  which  are  composed  of  many  small 
crystals  arranged  in  parallel  position  (Fig.  238).  The  angles 
between  the  branches  correspond  in  value  to  the  angles  between 
the  crystallographic  axes  of  the  little  individuals  or  to  some 
other  equally  characteristic  angles.  These  are  often  known  as 


FIG 

con  crystal  com- 
posed of  several 
individuals  a  r  - 
ranged  in  a  verti- 
cal pile. 


&x* 


FIG.  238. — Skeleton  crystals  of  argentite  in  section  parallel  to  cubic  face  (a). 
Portion  of  same  magnified  (&).  Showing  grouping  of  tiny  octahedrons  arranged  in 
lines  parallel  to  the  crystallographic  axes.  (After  Tschermak.) 

dendritic  growths.     They  occur  in  moss  agate  and  in  many  glassy 
volcanic  rocks. 

A  druse  is  a  crust  composed  of  small  crystals  implanted  side 
by  side  in  approximately  parallel  positions  upon  some  other 
mineral  or  upon  a  rock  surface  (Fig.  239). 


150  GEOMETRICAL   CRYSTALLOGRAPHY 

More  frequent  even  than  the  parallel  growth  of  similar  mineral 
species  is  the  case  of  parallel  growths  of  crystals  of  different  sub- 
stances, which  are,  however,  analogous  in  composition.  Sub- 
stances like  calcite  (CaCO3)  and  magnesite  (MgCO3)  possess 
similar  though  not  identical  crystal  forms.  Such  substances  are 
known  as  isomorphous  substances  (see  page  229).  These  often 
crystallize  together  so  that  the  axes  of  the  crystals  of  both  sub- 
stances are  approximately  parallel.  Often  the  crystallization  is 
so  intimate  that  the  different  substances  cannot  be  detected  even 
under  high  powers  of  the  microscope.  Calcite,  for  instance,  often 


FIG.  239 — Coating,  or  druse,  of  small  crystals  of  smithsonite  (ZnCO3)  on 
massive  smithsonite. 

contains  small  quantities  of  magnesium,  which  may  occur  in  the 
compound  as  intermixed  magnesite.  At  other  times  a  nucleus 
of  one  substance  may  be  surrounded  by  an  envelope  of  another 
substance,  and  this  in  turn  by  an  envelope  of  still  a  third  substance. 
A  cross  section  through  such  a  growth  will  show  a  nucleus  sur- 
rounded by  concentric  zones  of  varying  composition.  This  is 
known  as  a  zonal  growth.  It  is  well  exhibited  in  the  garnets, 
the  feldspars  (Fig.  240)  and  the  pyroxenes,  where  the  difference 
in  composition  of  the  different  zones  is  indicated  by  a  difference 
in  color  or  by  the  effect  of  the  different  layers  upon  polarized 
light. 

A  third  class  of  partial  parallelism  is  noted  in  the  association 
of  mineral  species  of  entirely  different  compositions,  possessing 


CRYSTAL    AGGREGATES 


entirely  different  crystal  forms.  Little  prismatic  crystals  of 
ruiile  (TiO2),  for  instance,  are  often  implanted  on  tabular  crystals 
of  hematite  (Fe2O3)  in  such  a  way  that  the  vertical  axes  of  the 
rutile  are  parallel  to  one  of  the  lateral  axes  of  hematite  (Fig.  241). 


FIG.  240. — Section  of  zonal  orthoclase  in  rock. 
Magnified  7^  diameters.      (After  Rosenbusch.) 

Many  other  instances  of  a  like  association  of  different  mineral 
species  are  known,  as,  for  instance,  pyrite  (isometric,  FeS2)  on 
marcasite  (orthorhombic  FeS2)  (Fig.  242)  and  albite  (NaAlSi3O8) 
on  orthoclase  (KAlSi3O8)  (Fig.  243). 


FIG.  241. — Small  crystals 
of  rutile  attached  to  crystal 
of  hematite. 


FIG.  242. — Little  crystals  of 
pyrite  implanted  on  marcasite. 
(After  Linck.} 


Twinned  Crystals. — Often  two  or  more  crystals  or  parts  of 
crystals  are  so  grouped  that  they  are  symmetrical  to  each  other 
with  respect  to  a  plane  between  them,  that  is  not  a  plane  of 
symmetry  for  either  individual.  These  groups,  or  twinned 
crystals  (Fig.  244),  are  of  great  importance  crystallographically, 


152  GEOMETRICAL    CRYSTALLOGRAPHY 

as  the  nature  of  the  twinning  is  very  characteristic  for  different 
substances. 

Twinning  Plane. — The  twinning  plane  is  the  plane  about 
which  the  twinned  individuals  are  symmetrical  (plane  ABCD  in 
Fig.  244) .  It  can  never  be  a  plane  of  symmetry  for  the  individuals, 
for  in  this  case  a  parallel  growth  would  result,  but  it  may  be  any 
other  crystallographic  plane  possible.  Usually  the  twinning 
plane  is  one  with  very  simple  indices. 

The  twinning  axis  is  a  line  about  which  one  of  the  twinned 
individuals  or  parts  of  individuals  may  be  supposed  to  be  revolved 
and  brought  into  a  parallel  position  with  the  other.  The  twin- 
ning axis  is  usually  normal  to  the  twinning  plane.  The  plane  of 


FIG.  243. — Crystals  of  albite 
on  orthoclase  with  vertical  axes 
of  both  parallel.  (After  Linck.} 


FIG.  244. — Contact 
twin  with  twinning 
plane  and  composi- 
tion faces  the  same, 
viz.:  ABCD. 


union  between  the  two  twinned  parts  is  the  composition  face,  and 
this  may  or  may  not  be  coincident  with  "the  twinning  plane.  In 
most  cases  of  twinned  crystals  there  is  observed  a  re-entrant 
angle  between  certain,  of  the  contiguous  planes  on  opposite  sides  of 
the  composition  face,  which  angle  serves  as  a  distinguishing  mark 
of  twinning  (see  top  of  crystal  in  Fig.  244). 

Contact  Twins. — When  parts  of  two  crystal  individuals  are 
united  in  a  plane  in  such  a  way  that  practically  all  of  each  indi- 
vidual is  on  one  side  only  of  the  plane,  the  twin  is  known  as  a 
contact  twin. 

In  figure  244,  which  represents  a  contact  twin  of  monoclinic 


CRYSTAL    AGGREGATES 


153 


gypsum  (CaSO4+2H2O),  the  twinning  plane  is  parallel  toooPoo, 
which  is  also  the  composition  plane.  In  albite,  which  is  triclinic 
(Fig.  245),  the  twinning  plane  is  ooPoo(M),  and  the  composition 
plane  the  same.  In  orthodase,  which  is  monoclinic  (Fig.  246),  the 
twinning  plane  is  oo  P  oo ,  and  the  composition  face  is  oo  P  oo  (M) 
(corresponding  to  ocPoo  in  albite). 

Penetration  Twins. — Frequently  the  twinning  individuals 
penetrate  each  other,  so  that  they  cannot  be  said  to  have  any 
composition  face — they  are  apparently  grown  through  one  an- 
other. Such  twins  are  known  as  penetration  twins. 


FIG.  245. 


FIG.  246. 


Twins  of  feldspars  possessing  different  twinning  planes  but  corresponding  com- 
position faces.  The  twinning  plane  in  Fig.  245  coincides  with  the  composition 
face.  In  Fig.  246  the  two  are  at  right  angles. 

Figure  [247  illustrates  an  interpenetration  twin  of  fluorite 
(CaFl2)  in  which  two  cubes  are  so  placed  that  they  are  sym- 
metrical about  a  plane  parallel  to  an  octahedral  face.  Figure 
248  is  an  interpenetration  twin  of  orthodase  with  the  same  law  of 
twinning  as  in  figure  246;  i.e.,  ooPoo  is  the  twinning  plane  and 
ocP  oo  the  composition  face. 

When  the  parts  of  such  twins  are  of  a  low  grade  of  symmetry 
and  by  their  intergrowth  tend  to  produce  a  body  with  a  higher 
grade  of  symmetry,  the  resulting  twin  is  known  as  a  supplementary 
twin.  Thus  positive  and  negative  hemihedrons  by  twinning  may 
produce  a  form  with  the  geometrical  symmetry  of  a  holohedron. 
Figure  249  represents  the  twinning  of  two  right-handed 
tetartohedral  crystals  of  quartz  reproducing  a  form  with 
hemihedral  symmetry. 


GEOMETRICAL   CRYSTALLOGRAPHY 


Repeated  Twins. — Although  the  term  twin  suggests  that 
twinned  crystals  consist  of  two  parts  in  the  twinning  relation  with 
respect  to  one  another,  it  nevertheless  often  happens  that  to  the 


FIG.  247 .  —  Interpenetra- 
tion  twin  consisting  of  two 
cubes  of  fluorite  twinned 
about  an  octahedral  plane. 


FIG.  248.  —  Interpenetra- 
tion  twin  of  feldspar  with 
same  twinning  plane  and 
composition  face  as  in  Fig. 
246. 


A  C  B 

FIG.  249.  —  Supplementary  twin  of  quartz  (C),  produced  by  combination  of  two 
right-hand  crystals  (A  and  B)  of  which  one  with  respect  to  the  other  is  revolved 
about  the  c  axis  through  an  angle  of  60°.  (After  Grotti). 


two  a  third,  a  fourth,  etc.,  are  added,  each  possessing  a  twinned 
relation  to  those  contiguous  to  it.  Thus  trillings,  fourlings,  etc., 
are  produced. 

Repeated  twinning  may  take  place  in  either  one^of  two  ways. 


CRYSTAL    AGGREGATES 


The  twinning  planes  between  the  contiguous  individuals  may  be 
parallel  to  each  other  or  they  may  not. 

In  the  former  case  the  alternate  individuals  are  in  parallel  posi- 
tion, as  is  indicated  in  figure  250,  which  represents  a  series  of  five 
orthorhombic  aragonite  (CaCO3)  plates,  twinned  about  parallel 
planes  of  oo  P.  A  cross  section  of  such  a  twin  is  pictured  in  figure  251. 
Twins  of  this  ^kind  are  known  as  polysynthetic  twins.     Each 
plate  is  called  a  lamella,  and  successive 
lamellae  are  in  reversed  positions  with 
respect  to  their  neighbors.     Fig.  252  is 
a  polysynthetic  twin  of  albite  twinned  in 
the  same  way  as  is  represented  in  Fig. 
245  (page  153). 


FIG.  250.  FIG.  251. 

Repeated  twin  of  aragonite  (Fig.   250)  and  cross  section  of  same  (Fig.   251) 
showing  successive  twinning  planes  parallel. 

Often  a  series  of  very  thin  twinned  lamellae  (Fig.  253)  produces 
a  striation  on  the  faces  of  crystals  or  of  cleavage  pieces.  This 
striation  is  well  exhibited  by  cleavage  pieces  of  triclinic  feldspar 
(Fig.-  254).  Light  will  be  reflected  in  the  same  direction  by 
alternate  lamellae,  so  that,  when  the  specimen  is  held  in  such  a  way 
as  to  catch  the  light  from  a  distinct  source,  one  set  of  lamellae 
will  appear  glistening  while  the  intermediate  lamellae  are  dull. 
Upon  turning  the  specimen  through  a  slight  angle  the  reflecting 
lamellae  become  dull  while  those  that  were  originally  dull  become 
bright.  This  phenomenon  is  the  oscillatory  twinning  referred 
to  on  page  136. 

When  the  twinning  planes  between  the  successive  individuals 
of  a  repeated  twin  are  alternate  planes  belonging  to  the  same 
form,  the  twin  turns  on  itself,  producing  a  circular  or  wheel-shaped 
group,  known  often  as  a  cyclic  twin. 


156 


GEOMETRICAL    CRYSTALLOGRAPHY 


Figure  255  is  an  illustration  of  an  aragoniie  trilling  in  which 
the  two  symmetrical  planes  of  ocP  become  successively  the 
twinning  plane.  By  further  repetition  of  this  twinning  a  nearly 


FIG.  252. — Polysynthetic 
twin  of  albite  with  thin 
lamella  in  center. 


FIG.  253. — Diagrammatic 
sketch  of  polysynthetic  twin 
of  albite  with  many  lamellae. 


complete  circle  or  hollow  cylinder  may  be  formed.  Figure  256 
is  the  cross  section  of  such  a  repeated  twin  in  which  four  individ- 
uals are  twinned. 

The   examples    of   repeated    twinning   given   above    are    all 


FIG.  254. — Twinning  striations  on  cleavage  surface  of  oligoclase.     The  bands 
are  due  to  the  alternation  of  lamellae  that  reflect  light  differently.     Natural  size. 

illustrations  of  contact  twins.  Interpenetration  twins  are  also 
repeatedly  twinned,  the  groups  taking  the  form  of  a  rayed  star 
or  of  a  bundle  of  symmetrical  plates. 

Figure  257  is  a  repeated  twin  of  cerussite  (PbCO3)  an  ortho- 


CRYSTAL    AGGREGATES 


157 


rhombic  mineral,  in  which  the  brachy-prism  oo  ?3  is  the  twinning 
plane;  and  figure  258,  a  twin  of  the  orthorhombic  chrysoberyl 
(BeAl2O4),  with  the  brachydome  Poo  the  twinning  plane. 
Figure  259  represents  a  very  complicated  interpenetration  twin 


\ 


FIG.  255. — Cyclic  trilling 
of  aragonite  with  successive 
twinning  planes  alternate 
oo  P  faces.  Compare  Fig. 

250. 


FIG.  256. — Cross  section  of  arago- 
nite fourling  with  successive  twinning 
planes  alternate  ooP  faces.  Com- 
pare Fig.  251. 


of  monoclinic  crystals  of  phillipsite  ((CaK2)Al2Si5O14  +  sH2O). 
The  three  columnar  portions  of  the  group  are  twinned  about  the 
same  plane  (A- A).  Each  column  is  composed  of  what  are 
apparently  two  individuals  twinned  about  another  plane  (B-B), 


FIG.  257. —  Repeated  in- 
terpenetration twin  of 
cerussite. 


FIG.  258. —  Repeated  in- 
terpenetration twin  of  chryso- 
beryl. 


but  each  of  these  seeming  individuals  is  in  reality  a  combination 
of  two  individuals  twinned  about  a  third  plane.  In  the  group 
there  are,  therefore,  twelve  individuals  twinned  according  to  three 
different  laws. 


158 


GEOMETRICAL    CRYSTALLOGRAPHY 


Mimicry. — By  the  repeated  twinning  of  crystals  of  a  low 
grade  of  symmetry  a  group  is  often  produced  which  appears  to  be  a 
simple  crystal  of  a  higher  grade  of  symmetry  than  that  of  its 


FIG.  259.  —  Complicated  inter- 
penetration  twin  'of  phillipsite. 
Twelve  individuals  twinned.  See 
text  for  explanation. 


FIG.  260. — Simple  crystal 
of  aragonite  (orthorhombic) 
with  coP(w),  <x>Pct>  (&)  and 
PS  (A). 


components.  Thus  the  orthorhombic  mineral  aragonite  (CaCO3) 
often  crystallizes  in  prismatic  crystals,  bounded  by  the  forms 
oo P  (m),  co Poo  (b),  and  Poo  (k)  (Fig.  260).  A  cross  section 


a  =  116  16 


FIG.  261. — Cross  section  of 
trilling  of  aragonite  with  twin- 
ning plane  alternate  ooP  faces. 
See  also  Fig.  256. 


FIG.  262. — Trilling  of  aragon- 
ite producing  a  group  that  re- 
sembles an  hexagonal  prism 
terminated  by  basal  planes.  A 
reentrant  angle  is  seen  on  the 
right.  Natural  size. 


through  a  single  crystal  has  the  outline  shown  in  one  of  the 
individuals  in  figure  261,  in  which  the  angle  between  the  ooP  faces 
is  116°  16'.  Trillings  formed  of  three  of  these  crystals  twinned 


CRYSTAL    AGGREGATES 


IS9 


parallel  to  ooP  have  a  cross  section  like  figure  261.  When  the 
spaces  between  the  dotted  lines  in  the  figure  and  the  body  of  the 
crystal  become  filled  with  mineral  substance,  the  trilling  strongly 
resembles  an  hexagonal  prism.  The  resemblance  is  made 
striking  by  the  close  approximation  of  the  angles  «  and  /?  to  the 
angles  of  the  hexagonal  prism  (120°).  The  angle  a=n6°  16' 
and  /?=i27°  28'.  Figure  262  is  the  reproduction  of  a  photo- 
graph of  such  a  trilling.  The  interpenetration  twin  of  chryso- 
beryl  (Fig.  255)  also  simulates  the  hexagonal  symmetry. 


CHAPTER  XIV. 
AMORPHOUS  SUBSTANCES  AND  PSEUDOMORPHS. 

Amorphous  Substances. — Although  the  great  majority  of 
chemical  compounds  possess  definite  forms,  there  are  some  to 
which  such  forms  seem  to  be  entirely  lacking.  Bodies  of  this 
kind  possess  neither  the  geometrical  properties  of  crystals  nor  do 
they  have  the  physical  properties  peculiar  to  crystallized  bodies. 
Their  internal  structure  has  not  the  regularity  of  that  of  crystals. 
Such  substances  are  said  to  be  amorphous,  or  they  are  described 
as  colloids.  Their  crystallizing  power  is  so  weak  that  it  is  not 
capable  of  causing  the  molecules  in  which  it  resides  to  group 
themselves  in  accordance  with  the  symmetry  of  any  crystal 
system,  except,  perhaps,  under  the  most  favorable  conditions. 
Under  ordinary  conditions  this  power  is  not  exerted  sufficiently 
to  effect  any  result,  and  so  the  material  is  put  together  according  to 
no  definite  plan,  and  consequently  it  possesses  no  definite  external 
form.  The  shapes  exhibited  by  such  substances  are  the  result 
of  external  conditions  or  of  forces  not  inherent  in  the  molecules 
of  the  substance.  They  are  largely,  if  not  entirely,  accidental. 

Pseudomorphs. — Idiomorphic  forms  of  crystals  are  deter- 
mined by  the  action  of  certain  forces,  called,  for  lack  of  a  more 
definite  name,  crystallizing  forces,  which  appear  to  be  inherent 
in  the  substance  of  which  the  crystals  are  composed.  The  forms 
produced  by  their  action  are  just  as  characteristic  of  the  crystal- 
lized material  as  are  its  chemical  reactions  with  other  substances. 
Very  frequently,  however,  substances  are  met  with  possessing 
definite  crystal  forms  that  are  different  from  those  which  they 
usually  possess,  but  which  are  similar  to  forms  possessed  by  some 
other  substance.  The  unusual  forms  originally  belonging  to 
some  pre-existing  substance  and  have  been  appropriated  by  the 
substance  now  possessing  them.  For  instance,  the  mineral  limonite 
(Fe4O3(OH)6)  is  usually  in  globular  or  botryoidal  forms  (see  Figs. 
229  and  230).  Sometimes,  however,  it  occurs  in  cubes  (Fig.  263), 

1 60 


AMORPHOUS    SUBSTANCES  AND   PSEUDOMORPHS  l6l 

which  are  known  to  be  the  forms  in  which  pyrite  (FeS2)  crystallizes. 
Bodies  possessing  forms  borrowed  from  other  substances  are  known 
as  pseudomorphs  (^ev&js,  false,  and  p>op<f>rj,  form).  This  term  is 
applied  not  only  to  the  form  itself,  but  as  well  to  the  substance 
exhibiting  it.  In  the  latter  sense  a  pseudomorph  is  a  body 
possessing  the  form  of  one  substance  and  the  chemical  and  physical 
properties  of  another.  Pseudomorphism  is  the  assumption  by  one 
substance  of  the  form  of  some  other  pre-existing  one. 

Explanation  of  Pseudomorphism. — The  explanation  of 
pseudomorphism  is  comparatively  easy.  A  substance  possessing 
its  own  distinctive  form  may  be  changed  by  the  action  of  the 


carbon  dioxide,  the  oxygen,  or  the  moisture  of  the  atmosphere, 
or  by  some  other  agency  into  another  substance  differing  from 
the  original  substance  in  nearly  all  of  its  morphological  and 
physical  properties.  The  material  of  the  original  substance  is 
completely  replaced  by  the  new  substance,  but  its  external  form 
remains  unchanged.  In  such  cases  there  results  a  pseudomorph. 
The  mineral  cuprite  (CuO)  often  forms  little  octahedra.  By 
exposure  to  the  atmosphere  cuprite  changes  to  malachite  (CuCO3. 
Cu(OH)2),  a  monoclinic  mineral  crystallizing  in  acicular  mono- 
clinic  crystals.  When  the  change  from  the  cuprite  to  the  mala- 
chite takes  place  slowly,  the  former  mineral  is  replaced,  molecule 
for  molecule,  by  the  latter,  the  result  being  a  mass  of  malachite 
with  the  outward  form  of  the  cuprite,  or  a  pseudomorph  of 
malachite  after  cuprite.  The  malachite  is  in  reality  monoclinic, 
as  may  be  learned  by  examining  it  optically,  but  it  possesses  the 
shape  of  a  regular  crystal.  In  the  same  way  we  find  gypsum 


162  GEOMETRICAL    CRYSTALLOGRAPHY 

(CaSO4  +2H2O)  pseudomorphs  after  anhydrite  (CaSO4),  limonite 
(Fe4O3(OH)6)  pseudomorphs  after  pyrite  (FeS2),  etc.  Petrified 
wood  may  likewise  be  described  as  a  pseudomorph  of  opal 
(SiO2  +Aq)  after  wood. 

Paramorphs. — Sometimes  what  is  apparently  the  same 
chemical  compound  occurs  in  two  different  forms  in  nature — it  is 
dimorphous,  crystallizing  in  one  form  under  certain  conditions  and 
in  an  entirely  different  form  under  other  conditions.  If,  after  the 
formation  of  crystals  of  such  a  substance,  the  conditions  change, 
the  entire  mass  of  the  crystal  may  pass  over  into  the  second  form, 
producing  a  pseumodorph  of  the  second  substance  after  the  first 
one.  The  external  form  of  the  second  substance  is  now  exactly 
similar  to  that  of  the  first  one,  but  its  molecular  structure  is 
entirely  different,  thus  giving  rise  to  a  genuine  pseudomorph  of  a 
kind  that  has  been  distinguished  by  the  name  paramorph.  A 
paramorph  is  a  pseudomorph  of  one  form  of  a  dimorphous  body 
after  the  other  form. 

One  of  the  most  familiar  illustrations  of  a  dimorphous  sub- 
stance is  sulphur,  which  separates  from  a  solution  in  carbon  bi- 
sulphide as  orthorhombic  crystals,  and  from  a  molten  mass  as 
monoclinic  needles.  When  allowed  to  stand  at  the  normal  tem- 
perature of  the  air,  the  monoclinic  variety  passes  into  the  ortho- 
rhombic  variety.  The  material  still  possesses  the  acicular  form  of 
the  monoclinic  sulphur,  but  its  molecular  structure  is  that  of  the 
orthorhombic  variety.  Here  we  have  an  example  of  a  paramorph 
of  orthorhombic  sulphur  after  the  monoclinic  variety. 

Two  Classes  of  Pseudomorphs. — The  processes  described 
above  as  originating  pseudomorphs  are  chemical,  hence  pseudo- 
morphs produced  by  them  are  known  as  chemical  pseudomorphs. 
There  is  another  class  of  pseudomorphs,  however,  known  as 
mechanical  pseudomorphs.  The  forms  of  these  bodies  are  not 
produced  by  the  replacement  of  the  substance  of  a  crystal,  par- 
ticle by  particle,  by  the  pseudomorphing  substance.  They  are 
produced  simply  by  the  filling  of  a  mould  left  by  the  solution  of 
some  pre-existing  crystal.  The  original  crystal  may  become 
incrusted  with  some  insoluble  material.  Its  substance  may  then 
be  dissolved,  leaving  a  cavity  of  the  shape  of  the  crystal.  If  this 


AMORPHOUS    SUBSTANCES  AND   PSEUDOMORPHS 


i63 


cavity  be  filled  with  a  new  substance,  and  then  the  enveloping 
material  be  removed,  the  new  substance  will  necessarily  possess 
the  form  of  the  original  crystal. 

At  Girgenti,  in  Sicily,  pseudomorphs  of  calcite  (CaCO3)  after 
sulphur  are  sometimes  met  with.  The  origin  is  explained  as 
follows:  The  sulphur  crystals  were  incrusted  with  a  coating  of 
barite  (BaSOJ.  The  temperature  in  the  neighborhood  rose  until 
the  sulphur  melted  and  disappeared,  leaving  a  mould  of  itself 
constructed  of  barite.  By  the  infiltration  of  a  solution  contain- 
ing calcium  carbonate  and  the  deposition  of  this  substance  as  cal- 


FIG.  264. — Fossils.     Pseudomorphs  of  dolomite  after  Mollusks  and  Coral. 

cite  in  the  mould,  the  cavity  was  filled.  Upon  the  removal  of  the 
barite  coating  a  mass  of  calcite  was  left  with  the  form  of  the  sul* 
phur  crystals. 

Fossilization. — Fossils  are  pseudomorphs  of  mineral  sub- 
stance after  organisms  or  parts  of  organism.  The  processes  of 
fossilization  are  exactly  analogous  to  those  of  pseudomorphism. 
The  original  organism  may  have  been  replaced,  molecule  by 
molecule,  with  the  fossilizing  substance,  or  it  may  have  been  dis- 
solved from  the  rock  in  which  it  was  imbedded,  leaving  a  cavity  of 
its  own  shape,  which  afterward  was  filled  with  mineral  substance. 
Fossils  produced  by  the  latter  process  preserve  only  the  external 
form  of  the  original  organism,  while  those  produced  by  replacement 
often  retain  even  the  minute  internal  structure  of  the  original. 

The  fossilizing  substance  is  usually  calcite  (CaCO3),  dolomite 
((CaMg)CO3),  or  silica  (SiO2)  in  some  form,  though  fossils 
composed  of  other  minerals  are  not  uncommon  (Fig.  264). 


CHAPTER  XV. 
CRYSTAL  PROJECTION. 

Projection. — By  the  term  projection  in  crystallography  is 
meant  the  representation  of  crystals  on  a  plane  surface;  i.e., 
on  the  surface  that  contains  their  figures.  There  are  several 
graphic  methods  by  which  the  planes  on  crystals  are  represented, 
among  which  are  the  linear  projection  and  the  spherical  pro- 
jection. These  exhibit  the  relations  of  the  planes  to  one  another 
without  reference  to  the  shape  of  the  crystal  bounded  by  them. 
Other  methods  of  projection — one  of  which  is  the  clinographic 
projection — represent  the  crystal  as  it  appears  to  the  eye  under 
certain  conditions.  This  projection  is  a  picture. 

The  Linear  Projection. — This  method  of  projection  repre- 
sents each  face  on  a  crystal  by  its  line  of  intersection  with  the  plane 
on  which  the  projection  is  made,  when  that  face  is  assumed  to  pass 
through  the  unity  of  the  vertical  axis.  The  plane  taken  as  the 
plane  of  projection  is  usually  that  which  includes  the  lateral  axes 
of  the  crystal.  The  projection  then  appears  as  a  number  of 
straight  lines  that  run  in  different  directions  across  this  plane 
and  at  different  distances  from  the  point  representing  the 
center  of  the  crystal.  Figure  265  shows  the  projection  of  the 
icositetrahedron. 

The  planes  are  all  assumed  to  pass  through  the  unity  on  the 
vertical  axis  because  the  intercepts  on  this  axis  cannot  be  indicated 
directly  in  the  projection.  They  must  therefore  be  indicated 
indirectly  and  this  is  done  by  imagining  them  moved  parallel 
until  they  pass  through  the  unity  on  c,  and  drawing  their  inter- 
sections with  the  plane  of  projection  by  joining  the  intercepts 
which  they  make  on  the  lateral  axes  in  this  position.  The  planes 
thus  projected  are  parallel  to  the  planes  on  the  crystal.  The 
ratios  of  their  intercepts  are  the  same  as  they  were  before  the 
imagined  movement,  and  consequently,  from  the  crystallographic 

164 


CRYSTAL   PROJECTION  165 

point  of  view,  they  are  the  same  planes.  Practically,  the  method 
consists  in  writing  the  intercepts  of  the  plane  to  be  projected  in 
the  form  of  a  ratio  and  reducing  to  unity  the  term  relating  to  the 
c  axis.  By  laying  off  on  the  lines  representing  the  lateral  axes 
distances  corresponding  to  the  reduced  intercepts  on  these  axes 
and  connecting  them  by  a  straight  line,  the  projection  of  the 
plane  is  obtained. 

In  all  the  systems  with  three  axes  the  lateral  axes  are  repre- 
sented in  the  projection  by  two  lines  perpendicular  to  each 
other.  They  are  usually  drawn  dotted  to  distinguish  them  from 


FIG.  265. — Form  2O2  and  its  linear  projection. 

the  projections  of  the  planes  which  are  drawn  solid.  In  the 
hexagonal  system  the  axes  are  represented  by  three  lines  inter- 
secting at  60°.  In  the  systems  with  equal  unities  on  the  lateral 
axes  equal  distances  are  laid  off  on  the  projection  of  these  axes  to 
represent  unity.  In  the  other  systems  the  distances  that  must  be 
laid  off  to  represent  the  unities  must  correspond  to  the  ratio 
between  the  unities  on  a  and  b  for  the  crystal  to  be  projected. 

Usually  only  the  upper  half  of  the  crystal  is  projected,  as  this 
projection  in  most  cases  represents  the  lower  half  as  well.  In  the 
case  of  certain  hemihedral  and  tetartohedral  forms,  in  which  the 
planes  on  the  two  halves  are  differently  related  to  the  axes,  two 
projections  must  be  made  to  represent  the  entire  form.  These, 
however,  may  be  indicated  in  the  same  figure  by  designating  the 


1 66 


GEOMETRICAL    CRYSTALLOGRAPHY 


upper  and  lower  planes  by  some  conventional  sign.  For  instance, 
the  projections  of  the  upper  planes  may  be  drawn  solid  and  those 
of  the  lower  planes  may  be  dotted. 

Suppose  the  form  262  is  to  be  projected.  The  lateral  axes 
are  first  indicated  as  two  dotted  lines  perpendicular  to  one  another 
(see  Fig.  265).  Convenient  equal  lengths  are  laid  off  as  unities. 
The  form  consists  of  12  planes  above  the  plane  of  projection, 
three  of  which  are  in  each  octant.  In  the  octant  in  which  all  the 
intercepts  are  positive  the  symbols  of  the  planes  are : 

20,  :  b  :  2C-,  a  :  2b  :  2c\  2a  :  2b  :  c,  or 

a  :  i/2b  :  c\  1/20,  :  b  :  c;  20,  :  2b  :  c,  when  the  intercept  on  c 
is  reduced  to  unity. 

For  the  projection  of  the  first  plane  lay  off  on  the  right-left 
axis  a  distance  equal  to  1/2  the  length  decided  upon  as  unity  and 
connect  this  point  by  a  straight  line  with  the  unity  distance  on 


FIG.  266. — Linear  projection  of  dodecahedron. 

the  front-back  axis  (A— A).  For  the  projection  of  the  second 
plane  lay  off  1/2  the  unity  distance  on  the  a  axis  and  connect  this 
by  a  straight  line  with  the  unity  point  on  b  (B— B).  For  the 
projection  of  the  third  plane  lay  off  two  unities  on  a  and  b  and 
connect  by  a  straight  line  (C— C).  By  a  similar  process  the 
projection  of  the  planes  in  the  other  three  upper  octants  are 
drawn  and  the  figure  is  completed  (see  Fig.  265). 

Planes  that  are  parallel  to  the  vertical  axis  must  necessarily 
pass  through  the  intersection  of  the  axes  in  the  projection,  since, 


CRYSTAL   PROJECTION 


i67 


when  such  planes  are  made  to  pass  through  the  unity  on  c, 
they  cut  this  axis  throughout  its  entire  length.  Thus  the  pro- 
jection of  oo  O  consists  of  two  lines  passing  through  the  unity 
distances  on  a  and  parallel  to  b,  two  lines  passing  through  the 


NJ 


P  ==    oP  x 

k  =  Co/'  55          c 
M=  GO/'C 


T  =- 


o   =    P 

w  =  2P  c). 


FIG.  267. — Crystal  of  orthoclase  with  linear  and  spherical  projections. 

unities  on  b  and  parallel  to  a,  and  two  lines  passing  through  the 
center  of  the  projection  and  bisecting  the  angles  between  the  axes. 
The  symbol  of  the  first  two  lines  is  a  :  oo  b  :  c;  of  the  second  pair, 
GO  a  :  b  :  c;  and  of  the  third  pair  a  :  b  :  ooc.  (See  Fig.  266.) 


1 68 


GEOMETRICAL    CRYSTALLOGRAPHY 


Spherical  Projection. — In  the  spherical  projection  the 
position  of  each  plane  on  the  upper  half  of  the  crystal  is  repre- 
sented by  the  point  of  intersection  of  a  perpendicular  to  the  center  of 
its  face  with  the  surface  of  a  hemisphere  at  whose  center  the  crystal 
is  supposed  to  be.  The  hemisphere  is  then  projected  on  a  plane 
passing  through  its  equator.  The  projection  appears  as  a  lot 
of  dots  arranged  with  the  same  symmetry  within  a  circle  as  that 
of  the  planes  on  the  crystal.  Figure  267  shows  the  projection 
of  a  crystal  of  orthoclase  (a  monoclinic  mineral)  by  the  linear 
and  the  spherical  methods. 

Crystal  Drawing. — The  representation  of  crystals  as  they 
appear  in  nature  (clinographic  projection),  or  the  drawing  of 
crystal  figures  such  as  are  used  in  this  book,  is  different  from 
ordinary  perspective  drawing  in  that  lines  which  are  parallel  on 
the  crystal  are  made  parallel  in  the  representation. 


FIG.  268. — Illustration  of  method  of  drawing  the  dodecahedron. 

In  constructing  a  crystal  drawing  the  axes  are  first  represented 
as  they  would  appear  if  the  eye  were  viewing  them  from  the  right 
and  from  above.  A  linear  projection  is  then  made  on  the  lateral 
axes,  and  lines  representing  the  directions  of  the  interfacial  edges 
between  contiguous  planes  are  drawn  from  the  ends  of  the  vertical 
axes  to  the  points  in  the  projection  where  the  lines  representing 
the  planes  intersect. 

Figure  268  shows  the  projection  of  the  axes  in  the  regular  sys- 
tem, with  the  projection  of  the  planes  of  the  dodecahedron  ( <x>  O) 
upon  the  plane  of  the  lateral  axes,  and  the  completed  drawing 
made  from  this  projection. 


CRYSTAL    PROJECTION 


169 


Projection  of  the  Crystal  Axes. — The  most  important 
step  in  the  drawing  of  any  crystal  is  the  projection  of  its 
axes,  because  these  serve  as  the  foundation  upon  which  the 
drawing  is  constructed.  After  the  preparation  of  the  per- 
spective view  of  the  axes  the  completion  of  the  drawing  is 
comparatively  easy. 

Different  methods  of  making  the  projection  of  the  axes  are  em- 
ployed, the  choice  between  them  depending  upon  those  features 
of  the  crystal  that  it  is  desired  to  emphasize.  For  general  pur- 
poses the  vertical  axis  is  repre- 
sented as  a  vertical  line,  and  the 
lateral  axes  are  drawn  in  perspec- 
tive on  the  assumption  that  they 
are  viewed  by  the  eye  at  a  certain 
angular  distance  (8)  to  the  right 
of  the  center  of  the  crystal  and 
elevated  a  certain  angle  (e)  above 
it.  Different  values  may  be 
assigned  to  8  and  £,  and  as  a 
result  different  views  of  the 
crystal  may  be  shown.  It  is 
usual,  however,  to  assume  such  values  as  may  be  expressed  by 
a  simple  ratio  between  equal  axes  after  projection.  If  the  ratio 
between  the  two  axes  OI  and  OK'  (Fig.  270)  is  i  :  3  and  the 
ratio  between  AI  and  OI  is  i  :  2,  then  8=  18°  26'  and  ^  =  9°  28'. 

Figure  269  represents  the  normal  position  of  the  lateral  axes 
in  the  isometric  or  tetragonal  system  as  viewed  from  the  top  of  the 
crystal;  i.e.,  along  the  direction  of  the  vertical  axis,  c.  If  viewed 
from  the  direction  of  the  axis  AA(a)  this  will  appear  as  a  point, 
while  BB(6)  will  appear  in  its  true  length. 

After  revolution  to  the  left  through  the  angle  AOA'  (  =  8= 
1 8°  26')  the  axis  AA  will  have  the  position  A' A';  i.e.,  its  front  half 
will  be  lengthened  to  OI,  and  BB  will  assume  the  position  B'B'; 
i.e.,  its  right  half  will  be  shortened  to  OH. 

If,  now,  the  eye  be  elevated  (at  the  angle  ?)  the  lines  A'l,  AO, 
and  B'H  will  be  projected  below  I,  O,  and  H  to  distances  propor- 
tional to  the  lengths  of  the  respective  lines.  If  8=18°  26'  and 


i  yo 


GEOMETRICAL    CRYSTALLOGRAPHY 


£  =  9°  28',  then  OI  :  OH=i  :  3  and  a'l  :  OI  as  i  :  2   (in  which 
a'l  is  the  projection  of  AI  below  I). 

Projection  of  the  Axes  for  the  Isometric  System. — If  the 
values  for  8  and  e  be  assumed  as  above,  then  the  method  of  con- 
struction of  the  isometric  axes  is  as  follows:  Draw  two  lines  LL' 

and  KK'  at  right  angles  to  one 
another  (Fig.  270).  Make  KO  = 
K'O  =  unity  on  b,  and  divide  KK' 
into  three  equal  parts.  Draw 
verticals  through  the  four  points 
thus  obtained  on  KK',  and  below 
K'  lay  off  K'H  =  1/2  K'O.  Draw 
HO,  which  will  give  the  direction 
of  the  front  lateral  axis.  Its 
length  will  be  that  portion  of  this 
line  included  between  the  two 
inner  verticals,  A  and  A'. 

Draw  AS  parallel  to  K'O  and 
connect  the  points  S  and  O. 
From  the  intersection  of  this  line 
with  the  inner  vertical,  T,  draw 
TB  parallel  to  K'K.  From  point,  B,  thus  obtained  draw  the 
line  BB'  through  O.  This  will  be  the  second  lateral  axis,  a. 

Below  K,  lay  off  KQ=i/3  OK  and  make  OC  =  OC'  =  OQ; 
then  CC'  will  be  the  length  of  the  vertical  axis.* 

Projection  of  the  Axes  for  the  Tetragonal  and  Ortho- 
rhombic  Systems. — The  axes  constructed  for  the  isometric  sys- 
tem may  be  readily  adapted  to  both  the  other  systems  with  rect- 
angular axes  by  merely  laying  off  portions  of  the  lines  AA'  and 

*  In  order  to  avoid  the  necessity  of  projecting  the  axes  each  time  a  drawing  is 
to  be  made,  it  is  advisable  to  construct  a  set  on  a  piece  of  cardboard  and  prick  holes 
at  the  ends  of  the  axes  and  their  point  of  intersection.  The  axes  can  then  be  trans- 
ferred to  drawing-paper  by  making  dots  through  these  holes  and  connecting  them 
by  straight  lines.  The  relative  lengths  of  equal  unity  distances  will  be  indicated  by 
the  positions  of  the  dots.  Longer  or  shorter  unity  distances  are  secured  by  increas- 
ing or  diminishing  proportionately  the  lengths  thus  obtained.  It  is  important  that 
the  construction  be  made  with  all  possible  accuracy,  otherwise  the  completed 
figures  may  be  distorted.  This  is  secured  in  part  by  making  the  original  drawing 
of  such  a  size  (e.g.,  by  making  the  entire  length  of  the  b  axis  =  4  inches)  that  small 
errors  will  be  practically  eliminated  when  the  lengths  of  the  axes  are  reduced  to  the 
dimensions  ordinarily  employed  in  drawing  crystals. 


FIG.  270. — Construction  of  isometric 
axes. 


CRYSTAL   PROJECTION 


171 


CC',  which  are  proportional  to  the  lengths  expressed  in  the  axial 
ratios  of  the  crystals  to  be  figured. 

For  instance,  if  the  axial  ratio  of  the  crystal  to  be  drawn  is 
a  :  b  :  c=i.$  :  i  :  2.4,  proceed  as  follows :  Transfer  the  perma- 
nent projection  of  the  axes  to  the  paper  upon  which  the  drawing 
is  to  be  made.  Take  proportional  lengths  of  the  axes  as  thus 
constructed  if  more  convenient  than  the  entire  lengths.  These 
distances  will  represent  ratios  of  i  :  i  :  i  on  the  three  axes.  In- 
crease the  length  on  a  by  .  5  of  itself 
and  that  on  c  by  i .  4  of  itself.  The 
resulting  lengths  will  have  the  rela- 
tions i .  5  :  i  :  2 . 4,  or  the  axial  ratio 
desired.  Indicate  these  by  dots 
and  treat  them  as  the  unity  inter- 
sections on  the  several  axes. 

In  the  case  of  a  tetragonal  crystal 
like  zircon,  the  axial  ratio  of  which 
is  a  :  c  :  :  i  :  .64,  the  two  lateral 
axes  remain  unchanged,  while  the 
vertical  axis  must  be  made  .64  of 
the  length  CC'. 

For  an  orthorhombic  crystal  the  axis  BB'  alone  remains  un- 
changed, while  AA'  and  CC'  are  both  changed  to  the  propor- 
tionate lengths  belonging  to  the  substance  in  question. 

Projection  of  theMonoclinic  Axes. — To  project  the  inclina- 
tion, /?,  of  the  clinoaxis,  construct  the  axes  as  in  the  isometric 
system,  and  then  lay  off  Oc=OC.cos  /?,  and  on  OA'  lay  off 
Oa  =  OA'.  sin  /5  (Fig.  271).  From  c  draw  a  line  parallel  to  OA', 
and  from  a  another  parallel  to  OC.  From  their  intersection,  a 
line  (DD')  drawn  through  O  will  give  the  direction  of  the  clino- 
axis. The  directions  of  the  other  two  axes  remain  unchanged. 
The  relative  lengths  of  the  axes  must  now  be  laid  off,  accord- 
ing to  the  axial  ratio  of  the  substance,  as  in  the  orthorhombic 
system. 

Projection  of  the  Triclinic  Axes. — In  this  case  all  three  axes 
of  reference  intersect  obliquely  b Ac=  <*,  a Ac  =  fi,  a A  6  =  7-.  If 
we  start  with  the  isometric  axes,  the  first  step  in  their  adaptation 


FIG.  271. — Construction  of  mono- 
clinic  axes. 


172 


GEOMETRICAL    CRYSTALLOGRAPHY 


D' 


FIG.  272. — Construction  of  triclinic 
axes. 


to  the  triclinic  system  is  to  obtain  the  direction  of  the  two  vertica 
axial  planes  or  pinacoids.  To  do  this,  lay  off  (Fig.  272)  on  OB 
Ob  =  OB.  sin  (/>  (</>  being  the  angle  ooPoc  A  ooP  ex,  which  is 

evidently  not  the  same  as  ?-),  and 
on  OA,  Oa=OA.  cos  <f>.  The 
line  drawn  from  the  angle  d  of 
the  parallelogram  adbO  through 
O  will  give  the  direction  of  the 
macropinacoidal  section,  DD'. 
To  obtain  the  direction  of  the 
macroaxis  (b),  lay  off  on  OD'> 
Od'  =  OD.  sin  a;  and  on  OC, 
Oc=OC.  cos  «.  From  the 
parallelogram,  d'OcK',  thus 
obtained,  the  diagonal,  K'K, 
gives  the  macroaxis.  In  a 
similar  manner,  the  brachyaxis 

(a),  HH',  is  found  by  laying  off  on  OA',  Oa'  =  OA.  sin  /?:  and 
upon  OC,  Oc'  =  OC.  cos  /?.  The  vertical  axis  CC'  and  the 
lateral  axes  HH'  and  KK'  thus  obtained  are  the  axes  of  a  triclinic 
crystal  in  which  a  :  b  :  c  = 
i  :  i  :  i.  Their  relative 
lengths  must  now  be  given 
them  in  accordance  with 
the  axial  ratio  of  the  sub- 
stance, just  as  in  the  ortho- 
rhombic  and  monoclinic 
systems. 

Projection  of  the 
Hexagonak  Axes. — Con- 
struct an  orthorhombic  set 
of  axes  whose  axial  ratio, 
a  :  b  :  c,  is>/3  (=1.732)  : 
i  :  c  (c  being  given  the 
value  of  the  vertical  axis  belonging  to  the  substance  to  be  drawn) 
(Fig.  273);  connect  the  extremities  of  the  two  lateral  axes, 
and,  in  the  rhomb  thus  formed,  the  obtuse  angles,  at  the  ends 


4-a, 


FIG.  273. — Construction  of  hexagonal  axes. 


CRYSTAL    PROJECTION  173 

of  the  b  axis,  will  be  exactly  120°.  If  .lines  be  now  drawn 
parallel  to  b,  through  points  on  the  axis  a,  half  way  between  its 
extremities  and  the  center  o,  the  rhomb  will  be  converted  into  a 
hexagon,  with  all  of  its  angles  exactly  120°.  If  we  connect  the 
diagonally  opposite  angles  of  this  hexagon,  we  shall  obtain  the 
projection  of  the  hexagonal  axes  required. 

Construction  of  the  Drawing. — Having  made  a  projec- 
tion of  the  axes  the  next  step  is  to  transfer  to  the  lateral  axes  the 
plane  projection  of  the  faces  on  the  crystal  to  be  drawn  (see  Fig. 
268).  Remembering  that  all  the  lines  in  this  projection  repre- 
sent the  traces  of  the  planes  when  they  are  supposed  to  pass 
through  the  unity  point  on  c,  it  follows  that  this  point  is  common  to 
all  the  planes  represented  in  the  projection. 

The  outline  of  the  crystal  is  shown  by  drawing  the  interfacial 
edges  between  neighboring  planes.  These  edges  are  represented 
as  lines  in  the  drawing.  It  is  only  necessary  ,then,  to  find  one  point 
other  than  the  unity  on  c  which  is  common  to  the  two  planes  whose 
intersection  is  desired.  This  is  the  point  of  intersection  of  the 
two  lines  representing  the  planes  in  the  projection.  Thus  we 
obtain  two  points  which  are  at  the  same  time  in  the  two  intersect- 
ing planes.  The  line  joining  them  gives  the  direction  of  their 
interfacial  edge.  If,  therefore,  we  draw  a  line  from  the  point 
representing  the  intersection  of  the  two  planes  in  the  projection 
to  the  unity  point  on  c  we  have  the  direction  of  the  interfacial  edge 
of  these  planes.  In  cases  when  the  intersecting  planes  are  repre- 
sented by  parallel  lines  in  the  projection,  their  interfacial  edge  is 
indicated  in  the  drawing  by  a  line  parallel  to  these  lines  in  the 
projection. 

It  will  be  noted  that  the  lines  drawn  by  this  procedure  indicate 
only  the  directions  of  the  desired  intersections.  They  all  diverge 
from  the  unity  point  on  the  vertical  axes.  In  the  crystal  the  inter- 
sections are  not  so  distributed.  They  are  lines  joining  certain 
points  on  the  crystal  surface  which  may  or  may  not  be  at  the 
unity  point  on  the  c  axis.  In  constructing  the  drawing  the  lines 
with  their  proper  directions  must  be  drawn  from  points  that  have 
the  relative  positions  of  the  corresponding  points  on  the  crystal. 

Usually  a  prominent  face  is  first  outlined  by  drawing  the 


74  GEOMETRICAL    CRYSTALLOGRAPHY 

proper  lines.  Then  from  points  on  the  outline  of  the  face  other 
lines  are  drawn  to  represent  the  interfacial  edges  that  extend  from 
it,  and  so  on  until  the  complete  figure  is  produced.  The  size  of  the 
figure  will  be  determined  by  the  size  of  the  first  face  outlined.  Its 
proportion  will  in  some  cases  be  fixed  by  the  symmetrical  develop- 
ment of  the  figure.  In  most  cases,  however,  the  proportions  must 
be  controlled  by  noting  the  proportional  lengths  of  the  different 
interfacial  edges  on  the  crystal  and  making  them  correspond  in  the 
figure.  In  other  words,  the  direction  of  the  lines  representing  the 
interfacial  edges  are  fixed  by  the  projection,  but  the  habits  of  the 
crystals  represented  are  indicated  by  the  relative  sizes  of  the  planes 
in  the  drawing  or  the  relative  lengths  of  the  lines  representing  their 
interfacial  edges. 


PART  II. 
PHYSICAL  CRYSTALLOGRAPHY. 


CHAPTER  XVI. 

INTRODUCTION:  PHYSICAL  SYMMETRY  AND  PHYSICAL 
AGENCIES. 

Physical  Symmetry. — The  material  of  which  crystals  are 
composed,  like  all  other  matter  existing  throughout  the  universe, 
is  made  known  to  our  senses  through  its  properties.  The  sub- 
stance of  crystals  obeys  the  same  laws  of  physics  that  govern  all 
other  matter,  but  these  laws  manifest  themselves  a  little  differently 
than  they  do  in  non-crystallized  bodies,  because  of  the  fact  that 
the  crystal  particles  are  put  together  in  a  definite  manner. 

In  amorphous  bodies  physical  forces  produce  exactly  similar 
effects  when  acting  in  different  directions,  provided  the  bodies 
are  homogeneous  throughout.  In  crystalline  bodies  the  case  is 
different.  In  these  the  effects  of  forces  vary  with  directions,  and 
the  variation  is  in  the  closest  accord  with  the  symmetry  of  the 
substance  acted  upon;  i.e.,  with  the  arrangement  of  its  component 
particles  as  expressed  in  the  symmetry  of  its  idiomorphic  forms. 
Every  geometrical  plane  of  symmetry  is  also  a  physical  one,  and  all 
geometrically  equivalent  directions  are  also  equivalent  with  respect 
to  physical  agents.  When  any  given  force  acts  upon  a  crystal 
it  will  produce  similar  effects  along  parallel  directions,  and 
effects  varying  in  degree  along  different  directions  which  are  not 
symmetrically  disposed  about  planes  and  axes  of  symmetry. 

Of  course  a  force  like  the  force  of  gravity  may  act  upon  a 
crystal  as  a  whole,  and  then  the  effect  of  symmetry  will  not  be 
apparent.  In  all  cases,  however,  where  the  force  acts  successively 
upon  different  particles  of  the  crystal  or  when  it  is  transmitted 
by  these  particles,  the  effect  of  symmetry  is  clearly  noticeable. 

When  considering  the  effects  of  forces  acting  upon  the  surfaces 
of  crystals  it  must  be  kept  in  mind  that  the  symmetry  of  the  planes 
bounding  a  crystal  is  something  quite  different  from  the  symmetry 
of  the  crystal  itself.  Only  those  crystallographic  planes  of 

177 


i78 


PHYSICAL   CRYSTALLOGRAPHY 


symmetry  that  are  perpendicular  to  the  bounding  planes  can  be 
planes  of  symmetry  for  these  surfaces.  It  is  impossible  to  con- 
ceive of  a  plane  surface  symmetrically  disposed  about  a  plane 
inclined  to  it. 

As  an  illustration  of  the  relation  existing  between  the  sym- 
metry of  planes  and  that  of  the  crystals  on  which  they  occur,  let 
us  consider  the  basal  planes  on  holohedral  forms  belonging  to  the 
different  crystallographic  systems  (see  Fig.  274).  The  top  plane  of 
the  cube  in  the  isometric  system  corresponds  to  the  basal  pinacoid 
in  the  other  systems.  In  the  isometric  system  this  plane  is  sym- 
metrical with  respect  to  the  four  planes  inclined  to  one  another 
at  angles  of  45°  and  intersecting  in  a  common  line,  which  is  the 


FiG.  274. — Diagrams  illustrating  symmetry  of  planes  at  terminations  of  the  vertical 
axes  in  holohedrons  of  the  different  systems. 

vertical  axis  of  the  cube  (a) .  In  the  hexagonal  system  the  basal 
plane  is  symmetrical  with  respect  to  six  similarly  intersecting 
planes  inclined  to  one  another  at  angles  of  60°  (b).  The  cor- 
responding face  in  the  tetragonal  system  is  similar  to  the  cubic 
face  in  its  symmetry.  In  the  orthorhombic  system  it  is  sym- 
metrical with  respect  to  two  planes  of  symmetry  perpendicular 
to  each  other  (c).  In  the  monoclinic  system  the  basal  plane  is 
traversed  by  a  single  plane  of  symmetry  (d),  and  in  the  triclinic 
system  it  is  crossed  by  no  symmetry  plane  (e). 

Forces  acting  upon  the  faces  of  a  crystal  will  be  governed  in 
their  effects  by  the  symmetry  of  the  faces.  .From  the  symmetry 
of  the  effects  produced  the  symmetry  of  the  faces  is  disclosed  and 
from  the  symmetry  of  several  faces  the 'symmetry  of  the  whole 
form  to  which  they  belong  may  be  deduced.  Consequently, 
the  grade  of  symmetry  possessed  by  a  substance  may  be  dis- 
covered even  when  complete  crystals  are  not  obtainable. 

Physical  Agencies. — The  physical  forces  producing  effects 


PHYSICAL   SYMMETRY  AND   PHYSICAL  AGENCIES  179 

that  are  of  the  greatest  importance  in  determining  the  grade  of 
symmetry  of  crystals  may  be  classified  as  mechanical,  optical, 
thermal,  and  electrical.  The  nature  of  these  forces  is  a  subject 
dealt  with  in  physics.  At  present  we  are  more  directly  concerned 
with  the  character  and  distribution  of  the  effects  they  produce. 
These  effects,  as  has  already  been  stated,  accord  in  their  general 
character  with  the  symmetry  of  the  crystal  upon  which  they  act. 
The  relations  of  the  substance  toward  the  forces  are  known 
as  their  properties.  Thus  we  speak  of  the  optical  properties 
of  crystals  when  we  refer  to  their  action  with  reference  to  the 
forces  that  produce  light,  of  their  thermal  properties  when  we 
refer  to  their  action  under  the  influence  of  thermal  forces,  etc. 


CHAPTER  XVII. 
MECHANICAL  PROPERTIES  OF  CRYSTALS. 

Mechanical  Properties. — The  mechanical  properties  of 
crystals  are  those  they  exhibit  with  respect  to  mechanical  forces. 
They  are  discussed  under  the  following  heads:  Elasticity, 
tenacity,  cohesion,  cleavage,  fracture,  hardness,  and  density. 

Elasticity. — By  elasticity  is  meant  that  property  which 
causes  bodies  to  resist  forces  tending  to  change  their  form. 
This  power  of  resistance  varies  in  different  substances  and  varies 
along  different  directions  in  the  same  body,  provided  it  is  not 
amorphous.  It  is  expressed  by  the  coefficient  of  elasticity,  which 
is  the  relation  existing  between  the  length  of  a  standard-sized  bar 
of  the  substance  and  the  elongation  it  suffers  under  the  influence 
of  a  given  pull.  For  all  amorphous  bodies  the  coefficient  of  elasticity 
is  equal  in  all  directions;  that  is,  bars  of  equal  size  cut  from 
amorphous  bodies  in  any  direction  will  be  equally  elongated 
when  subjected  to  the  same  strain. 

In  crystals  the  value  of  the  coefficient  of  elasticity  varies  according 
to  the  direction  in  which  the  stress  is  applied.  In  isometric  crystals 
it  is  equal  in  directions  parallel  to  the  three  crystallographic  axes, 
and  varies  from  this  in  other  directions  always  being  equal,  how- 
ever, in  parallel  directions,  and  in  directions  that  are  symmetrical 
with  respect  to  one  another. 

The  coefficient  of  elasticity  in  salt  (NaCl),  for  instance,  is  as  i 
to  .7  in  rods  cut  perpendicular  to  the  cubic  face  and  those  cut 
perpendicular  to  the  octahedral  face. 

Although  the  symmetry  of  a  crystallized  body  may  be  deter- 
mined by  a  study  of  its  elastic  properties  even  when  crystals  of  it 
are  not  obtainable,  nevertheless,  since  there  are  much  more  con- 
venient methods  than  this  that  may  be  employed  for  the  purpose, 
the  elastic  properties  are  not  made  use  of  to  any  great  extent. 

Tenacity. — If  the  elasticity  of  a  body  is  defined  as  the  quan- 

180 


MECHANICAL    PROPERTIES    OF    CRYSTALS  l8l 

tity  of  resistance  it  opposes  to  deforming  influences,  we  may  de- 
fine tenacity  as  the  quality  of  this  resistance. 

With  respect  to  tenacity,  substances  are  distinguished  as  brittle, 
sectile,  malleable,  flexible,  and  elastic.  A  brittle  substance  is  one 
that  breaks  into  powder  when  cut  with  a  knife,  as  does  calcite 
(CaCO3).  A  sectile  substance  may  be  cut,  but  it  pulverizes 
under  blows,  as,  for  instance,  gypsum  (CaSO4 +2H2O).  A 
malleable  substance  flattens  when  hammered  upon,  as  copper  and 
other  metals.  A  flexible  substance  will  bend  when  subjected  to 
forces  properly  applied,  and  will  remain  bent  when  the  action  of  the 


FIG.  275. — Cleavage  piece  of  calcite  showing  cleavage  cracks. 

forces  ceases,  as  talc  (H2Mg3(SiO3)4)  and  asbestus  ((CaMg)SiO3). 
An  elastic  substance  will  fly  back  into  its  original  position  when  the 
force  that  bends  it  is  removed,  as  mica  (H2(KNa)Al3(SiO4)3). 

Cohesion. — Cohesion  is  the  resistance  offered  by  bodies  to 
the  separation  of  their  particles.  Those  special  characters  that 
are  dependent  upon  the  strength  of  this  resistance  are  cleavage, 
fracture,  and  hardness. 

Cleavage. — Many  crystals  possess  a  marked  tendency  to  split 
along  certain  directions  in  preference  to  others,  in  consequence  of 
differences  in  cohesive  power  in  different  directions.  The  planes 
along  which  such  splitting  occurs  are  known  as  cleavage  planes 
(Fig.  275).  They  must  be  perpendicular  to  the  direction  of  mini- 
mum cohesion  and  their  perfection  must  depend  upon  the  dif- 


182  PHYSICAL   CRYSTALLOGRAPHY 

ference  in  the  cohesive  force  along  different  directions — the 
greater  the  cohesion  difference  the  better  the  cleavage. 

The  cleavage  planes  are  always  parallel  to  planes  that  are 
crystallographically  possible;  that  is,  they  are  planes  with  rational 
indices.  Moreover,  if  one  crystal  face  is  parallelized  by  a  cleavage 
plane,  all  other  faces  belonging  to  the  same  crystal  form  are  also 
parallelized  by  cleavage  planes,  and  along  all  of  these  planes  the 
cleavage  is  equally  easy.  This  must  be  so  because  cohesion  is  a 
property  of  the  molecules  and  in  crystals  the  molecules  are  regu- 
larly arranged.  Their  geometrical  forms  and  physical  properties 
are  but  different  expressions  of  this  arrangement.  When  two 
different  cleavages  are  present,  i.e.,  when  a  crystal  cleaves  parallel  to 
faces  belonging  to  different  crystal  forms,  the  ease  with  which  the 
cleavages  are  produced  is  unequal,  and  the  character  of  the  surfaces 
produced  differs.  In  galena  (PbS),  for  instance,  cleavage  is  equally 
easy  in  three  directions  perpendicular  to  one  another;  i.e.,  in 
directions  parallel  to  cubic  faces — the  cleavage  is  cubical.  In 
sphalerite  (isometric  ZnS)  the  cleavage  is  equal  along  planes 
parallel  to  dodecahedral  faces — it  is  dodecahedral.  In  barite 
(orthorhombic  BaSO4),  on  the  other  hand,  there  are  two  unequal 
cleavages.  The  most  easy  one  is-  parallel  to  oP  and  the  most 
difficult  one  parallel  to  P. 

"The  differences  in  the  character  of  cleavages  produced  in  a 
substance  often  serves  to  determine  its  system  of  crystallization. 
Anhydrite  (CaSOJ,  for  instance,  cleaves  along  three  planes  per- 
pendicular to  one  another,  but  with  different  degrees  of  perfec- 
tion. The  differences  in  ease  with  which  the  cleavage  takes  place 
along  the  different  planes  serves  to  fix  the  symmetry  of  the  sub- 
stance as  orthorhombic. 

According  to  the  ease  with  which  cleavage  is  effected  and  the 
evenness  of  the  surfaces  produced  by  the  cleavage,  this  is  said  to  be 
very  perfect,  perfect,  distinct,  indistinct,  imperfect,  interrupted,  or 
difficult. 

The  relative  ease  with  which  cleavage  is  produced  along  dif- 
ferent planes  is  measured  by  cutting  rods  of  a  substance  with  their 
long  axes  parallel  to  different  crystallographic  lines,  and  then 
r  training  them  until  they  break.  The  symmetry  of  the  cleavage 


MECHANICAL    PROPERTIES    OF    CRYSTALS  183 

may  be  determined  by  measuring  the  force  of  the  strain  under 
which  the  cleavage  is  effected  in  a  number  of  rods. 

Gliding. — Many  crystals  possess  a  property  which  is  analo- 
gous to  shearing,  but  which  differs  from  this  in  the  fact  that  it 
occurs  only  along  certain  planes,  which  are  perpendicular  to  the 
direction  of  maximum  cohesion — and  which  are  known  as  glid- 
ing planes.  Small  portions  of  the  crystals  may  be  moved  along 
the  gliding  planes  without  being  separated  from  the  unmoved 


FIG.  276. — Diagrams  illustrating  shearing  of  ice.     (After  Linck.) 

A.  Rod  cut  from  ice  crystal  parallel  to  its  vertical  axis  (cc). 

B.  The  same  rod  after  being  bent  by  loading  its  center.     The  vertical  axis  in 
the  bent  portion  remains  parallel  to  its  original  position. 

C.  Explanation  of  method  of  bending.     Thin  slices  of  the  ice  shear  downward. 

parts.  In  many  instances  the  movement  consists  of  a  slipping 
along  a  series  of  parallel  planes  and  is  unlimited  in  amount,  thus 
resembling  true  shearing.  In  other  cases  the  movement  is  caused 
by  a  rotation  of  the  molecules  in  a  series  of  rows  parallel  to  the 
gliding  plane.  Each  molecule  moves  but  a  slight  amount  with 
reference  to  neighboring  molecules,  and  the  limit  of  movement  is 
reached  when  the  part  of  the  crystal  that  has  been  deformed  occu- 
pies the  twinned  relation  to  the  undeformed  part  with  the  gliding 
plane  as  the  twinning  plane. 

The  two  types  of  gliding  are  well  illustrated  by  ice  and  calcite 
(Fig.  276).     If  a  bar  cut  from  a  crystal  of  ice  parallel  to  its  vertical 


184  PHYSICAL    CRYSTALLOGRAPHY 

axis  be  placed  in  a  horizontal  position  and  supported  at  its  ends 
while  weighted  at  its  center,  the  middle  portion  of  the  bar  will 
slowly  sag  downward.  Examination  of  the  bent  portion  will  show 
that  the  sagging  is  not  due  to  bending,  as  the  crystallographic 
axis  will  be  found  to  have  suffered  no  deformation  in  the  sagged 
portion  of  the  bar.  It  will  still  remain  parallel  to  its  original 

position.  The  sagging  must  therefore 
be  due  to  downward  slipping  of  a 
large  number  of  parallel  lamellae  (see 
Fig.  276). 

If,  on  the  other  hand,  a  sharp  knife- 

__  blade  is  placed  perpendicular  against 

FIG.  277.  —  Artificial  twin  of     the  blunt  edge  of  a  cleavage  rhomb 

te  produce, 


substance,  a  slice  of  the  material  will  move  without  fracturing 
until  it  assumes  a  twinned  position  with  reference  to  the  rest 
of  the  calcite  (see  Fig.  277).  The  corresponding  axes  in 
the  moved  portion  will  no  longer  be  parallel  to  their  original 
positions,  but  they  will  be  rotated  into  twinned  positions.  The 
plane  between  that  portion  of  the  calcite  that  has  been  moved 
and  that  portion  which  has  not  been  disturbed  is  the  gliding  plane. 


o 


FIG.  278. — Diagrams  illustrating  process  of  gliding  in  calcite.     The  circle  drawn 
near  the  upper  corner  (A)  is  elongated  into  an  ellipse  after  gliding  (5). 

It  is  also  the  twinning  plane.  The  fact  that  the  horizontal  rows 
of  molecules  have  rotated  is  shown  by  the  change  of  circles  into 
ellipses  during  the  movement  (see  Fig.  278). 

Secondary  Twinning. — Very  frequently  gliding  is  produced 
in  opposite  directions  along  a  succession  of  parallel  planes  so 
that  there  results  a  series  of  lamellae  each  in  the  twinning  position 


MECHANICAL   PROPERTIES    OF    CRYSTALS  185 

with  respect  to  its  neighbors.  Polysynthetic  twinning  produced 
in  this  way  is  known  as  secondary  twinning  because  brought 
about  after  the  crystals  exhibiting  it  were  formed.  It  is  a  common 
phenomenon  in  the  calcite  grains  composing  marble  (see  Fig.  279). 
Percussion  Figures. — When  a  hard  point  is  placed  against 
the  face  of  a  crystal  and  then  is  tapped  with  a  sharp  stroke,  cracks 
may  be  produced  forming  a  star-like  figure,  the  shape  of  which 


FIG.  279.— Thin  section  of  marble  viewed  in  polarized  light.  The  dark  bars  are 
secondary  twinning  lamella?  due  to  gliding  under  the  influence  of  pressure.  Mag- 
nified about  5  diameters. 

is  characteristic  for  many  substances.  The  cracks  are  partings 
along  definite  crystal  planes,  and  are  closely  related  to  the 
gliding  planes. 

The  percussion  figure  on  the  cubic  face  of  halite  (NaCl) 
is  a  four-rayed  star  with  the  rays  parallel  to  the  diagonals  of  the 
cubic  face.  The  rays  are  cracks  that  are  perpendicular  to  the 
face  and  therefore  parallel  to  the  four  dodecahedral  planes  that 
truncate  the  vertical  edges  of  the  cube.  On  the  octahedral  face 
the  percussion  figure  is  three-rayed. 

A  six -rayed  star  is  produced  in  a  similar  manner  on  cleavage 
pieces  of  mica  (Fig.  280)  which  crystallizes  in  the  monoclinic 
system.  The  ray  which  is  parallel  to  the  plane  of  symmetry— 
the  clinopinacoid — is  larger  than  the  others,  and  is  called  the 


i86 


PHYSICAL   CRYSTALLOGRAPHY 


characteristic  ray.  By  its  means  the  position  of  the  clinopinacoid 
may  be  determined  in  plates  of  mica  that  show  no  crystal  planes. 

Pressure  Figures. — If  a  very  thin  cleavage  plate  of  mica  is 
placed  on  a  yielding  support  and  pressed  by  a  blunt  point,  another 
figure  is  produced  which  is  also  six-rayed  when  perfect  (Fig.  281). 
Usually,  however,  some  of  the  rays  are  missing  and  the  figures 
consist  of  three  or  four  cracks  only.  The  rays  of  the  pressure 
figure  are  perpendicular  to  those  of  the  percussion  figure. 

Cracks  having  the  directions  of  the  rays  of  the  pressure  figure 
are  often  observed  in  pieces  of  mica  and  sometimes  triangular 


FIG.  280. — Percussion  figure 
on  basal  plane  of  muscovite. 
The  mineral  is  monoclinic  and 
the  long  ray  is  parallel  to  the 
plane  of  symmetry,  ocPoo  . 


FIG.  281. — Pressure  figure  in 
basal  plane  of  muscovite.  The 
rays  bisect  the  angles  between 
those  of  the  percussion  figure. 


fragments  of  the  mineral  are  found  in  which  actual  separation  has 
occurred  along  these  directions  (Fig.  282).  The  phenomena  are 
due  to  pressure  exerted  on  the  crystal  substance  while  in  the 
rocks. 

Parting.— Regular  breaking  along  planes  which  are  not 
cleavage  planes  is  known  as  parting.  It  differs  from  cleavage  in 
that  it  occurs  only  in  certain  places;  i.e.,  along  the  cracks  produced 
by  pressure,  along  the  planes  separating  twinned  lamellae,  etc., 
while  cleavage  may  take  place  equally  well  anywhere  parallel 
to  the  cleavage  plane  (see  Fig.  282). 

Fracture. — When  a  force  breaks  a  crystal  in  a  direction  which 
is  neither  a  cleavage  nor  a  gliding  plane,  or  produces  a  break  in  an 
amorphous  body,  the  separation  takes  place  in  an  irregular  way. 


MECHANICAL   PROPERTIES    OF    CRYSTALS 


i87 


This  kind  of  breaking  is  known  as  fracture.  It  is  described 
according  to  the  character  of  the  surfaces  produced,  as  even, 
splintery,  earthy,  hackly,  or  conchoidal.  A  hackly  fracture  leaves 
surfaces  that  are  ragged  and  rough,  such  as  is  exhibited  by  a 
broken  piece  of  malleable  metal.  A  conchoidal  fracture  leaves 
surfaces  marked  by  concentric,  or 
nearly  concentric,  curved  lines,  like 
the  lines  on  many  shells  of  molluscs 
(see  Fig.  283).  This  kind  of  frac- 
ture is  best  exhibited  by  glass.  The 
names  applied  to  the  other  kinds 
of  fracture  are  self-descriptive. 

Hardness. — The  hardness  of  a 
substance  may  be  measured  in  a 
number  of  different  ways,  but  the 
results  obtained  are  not  comparable.  FlG •  28 2. -Fragment  of  muscovite 

parted  along  the  planes  of  the  pres- 

Consequently,  a  satisfactory  defini-    sure  figure.    (After  Linck.} 
tion  of  hardness  is  not  yet  possible. 

A  harder  substance  will  scratch  a  softer  one.     The  miner- 


FIG.  283. — Conchoidal  fracture  in  obsidian. 

alogist  Mohs  proposed  the  names  of  the  following  ten  minerals 
to  serve  as  a  scale  to  which  to  refer  all  other  minerals  with  respect 
to  hardness.  The  scale  begins  with  a  soft  mineral  and  ends 
with  the  hardest  substance  known. 


1 88 


PHYSICAL   CRYSTALLOGRAPHY 


MOHS'S    SCALE    OF   HARDNESS? 


i-Talc  (H2Mg3(Si03)4) 
2— Gypsum  (CaSO4+2H2O) 
3—Calcite  (CaCO3) 
4— Fluorite  (CaF2) 


6— Feldspar  (KAlSi3O8) 
7— Quartz  (SiO2) 
8— Topaz  (Al2F2SiO5) 
9 — Corundum  (A12O3) 


5— Apatite  (Ca5(PO4)3F)  10— Diamond  (C). 

A  mineral  that  neither  scratches  any  given  mineral  in  the  scale 
of  hardness  nor  is  scratched  by  it  is  said  to  have  the  same  hardness 
as  this;  if  it  scratches  one  of  the  scale  minerals  and  is  scratched 
by  the  next  harder  one,  its  hardness  is  between  that  of  the  former 
and  that  of  the  latter.  For  example,  a  mineral  that  neither 
scratches  quartz  nor  is  scratched  by  it  has  a  hardness  of  7. 
One  that  scratches  feldspar  and  is  scratched  by  quartz  possesses 
a  degree  of  hardness  between  6  and  7. 

Minerals  with  a  hardness  of  2  or  under  can  be  stratched  with 
the  finger-nail.  Those  whose  hardness  is  between  2  and  4  can 
be  scratched  easily  with  the  point  of  a  knife.  Those  with  a 
hardness  of  4  to  5  cannot  be  scratched  with  a  knife,  but  can 
easily  be  scratched  with  a  good  file.  Only  those  minerals  whose 
hardness  is  greater  than  5  will  scratch  window-glass.  Only  those 
with  a  hardness  above  7  will  strike  fire  with  steel. 

The  degrees  in  the  Mohs's  scale  are  entirely  arbitrary  and  the 
intervals  between  them  are  very  unequal.  The  following  table 
records  the  relative  hardness  of  the  minerals  comprising  the  Mohs 
scale  as  determined  by  different  men  using  different  methods: 

RELATIVE  HARDNESS    OF   CERTAIN   MINERALS  AS   DETERMINED   BY 
DIFFERENT   INVESTIGATORS. 


Mohs 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

Franz  

!3-5 

54 

'35 

390 

670 

54o 

1000 

Pfaff  

I  .IT. 

12  .03 

1C  .  -2 

77  .  T. 

Cl  .  C 

IQI 

2<4 

4  en 

IOOO 

Rosiwal  

•33 

1-25 

4-5 

s 

6.5 

37 

120 

T75 

TOOO 

140,000 

Jaggar  

.04 

.26 

•75 

1.23 

25 

40 

!52 

IOOO 

MECHANICAL   PROPERTIES    OF    CRYSTALS 


189 


Usual  Method  of  Determining  Hardness. — The  relative 
hardness  of  different  substances  is  usually  determined  by  measur- 
ing the  force  required  to  scratch  their  smooth  surfaces.  This 
is  done  by  means  of  an  instrument  known  as  a  ' skier ometer. 
One  form  consists  of  a  sharp  point  of  steel  or  diamond  capable 
of  being  weighted  and  a  movable  platform  that  can  be  drawn 
beneath  it.  A  plate  of  the  substance  to  be  tested  is  placed  on  the 


FIG.  284. — Sklerometer — an  instrument  for  determining  differences  in  hardness 
in  different  directions  on  crystal  faces. 

platform  and  the  point  pressed  down  upon  it.  The  force  neces- 
sary to  drive  the  point  sufficiently  deep  into  the  substance  to 
leave  on  it  a  scratch  when  the  platform  is  moved  is  a  measure  of 
its  hardness.  One  form  of  the  sklerometer  is  shown  in  the 
accompanying  figure  (Fig.  284).  Its  essential  features  are  the 
little  car  or  movable  platform  (k),  the  point  (m),  the  platform 
for  the  load  (p),  and  the  basket  (g),  containing  a  constant  weight 
by  which  the  car  is  drawn  beneath  the  point. 

Differences  in  Hardness.— As  the  result  of  experimentation 
with  the  sklerometer  it  is  learned  that  not  only  do  different  sub- 
stances possess  different  degrees  of  hardness,  but  also  that  the 


1 90  PHYSICAL   CRYSTALLOGRAPHY 

same  substance  when  crystallized  offers  greater  opposition  to  a 
scratching  agent  acting  in  a  certain  direction  than  to  the  same  agent 
acting  along  some  other  direction. 

The  triclinic  mineral  cyanite  (Al2SiO5),  for  instance,  resists 
scratching  parallel  to  c  much  less  effectually  than  scratching  at 
right  angles  to  this  direction.  Cyanite  is  harder  in  directions 
perpendicular  to  c  than  in  directions  parallel  to  this  axis.  In 
general  it  is  found  that  crystals  are  equally  hard  in  directions  that 
are  symmetrical  with  respect  to  each  other,  and  that  the  hardness 
differs  in  directions  that  are  not  symmetrical. 

The  differences  in  hardness  in  any  crystal  seem  to  be  governed 

largely  by  the  cleavage,  scratches  always  being  produced  more 

easily  parallel  to  the  cleavage  than  across 

\\\\\\\\\\\       it.     Since  the  cleavage  is  in  accord  with 

\\\\\\\\\\\  the  symmetry  of  the  crystal,  its  hardness 
must  likewise  be  symmetrical.  More- 
over, if  the  cleavage  is  inclined  to  the 

surface  under  investigation  the  hardness  varies  in  opposite 
directions.  In  figure  285,  for  instance,  the  hardness  is  greater 
in  the  direction  b  to  a  than  in  the  direction  a  to  b. 

Curves  of  Hardness. — A  curve  drawn  on  a  plane  surface 
in  such  a  way  as  to  express  the  differences  in  hardness  exhibited 
by  a  crystal  face  in  different  directions  is  known  as  a  curve  of 
hardness.  The  relative  hardness  in  different  directions  is 
represented  by  the  relative  lengths  of  straight  lines  passing  through 
the  center  of  the  curve  and  terminating  at  both  ends  in  its  cir- 
cumference. 

Figures  286  and  287  are  the  curves  of  hardness  for  the  cubic 
faces  of  crystals  of  halite  (NaCl)  and  of  fluorite  (CaF2),  respect- 
ively. In  the  first  figure  the  lines  along  the  diagonals  of  the 
faces  are  longer  than  those  parallel  to  their  edges.  This  indicates 
that  the  hardness  in  the  former  direction  is  greater  than  that  in 
the  latter.  The  hardness  along  intermediate  directions  is  inter- 
mediate in  degree.  In  the  case  of  fluorite  the  hardness  is  least 
in  directions  parallel  to  the  diagonals  of  the  faces  and  greatest 
in  directions  perpendicular  to  the  cubic  edges. 

The  symmetry  of  curves  of  hardness  always  corresponds  to  the 


MECHANICAL   PROPERTIES    OF   CRYSTALS  191 

symmetry  of  the  faces  for  which  they  are  constructed.  Con- 
sequently from  the  differences  in  hardness  exhibited  by  sub- 
stances the  symmetry  of  their  crystals  may  be  determined. 


FIG.  286.  FIG.  287. 

Curves  of  hardness  in  cubic  faces  of  halite  (Fig.  286)  and  fluorite  (Fig.  287.) 

Figure  288  shows  the  curve  of  hardness  as  determined  by 
three  observers  on  the  basal  plane  of  calcite,  which  is  an  hexagonal 
mineral  crystallizing  in  rhombohedral  hemihedral  forms.  Three 
planes  of  symmetry  are  perpendicular  to  the  basal  plane  and 


FIG.  288. — Curves  of  hardness  on  the  basal  plane  of  calcite  as  determined  by 
three  different  investigators.  Although  varying  in  shape  they  all  exhibit  the  same 
symmetry.  (After  Mutter}. 

these  intersect  one  another  at  angles  of  120°.  In  the  figure  their 
positions  are  indicated  by  the  lines  drawn  from  the  angles  of 
the  circumscribing  triangle  to  the  centers  of  the  opposite  sides. 


192  PHYSICAL    CRYSTALLOGRAPHY 

The  three  curves,  though  of  different  shapes,  are  all  symmetrical 
with  respect  to  these  three  lines.  Had  the  mineral  been  holo- 
hedral  the  curves  would  have  been  symmetrical  about  six  lines 
intersecting  at  60°. 

Density  or  Specific  Gravity.— The  density  or  specific 
gravity  of  a  body  is  the  relation  existing  between  its  weight  and 
the  weight  of  an  equal  volume  of  water  at  a  given  temperature. 
In  brief,  the  specific  gravity  of  a  substance  is  the  weight  of  a 
cubic  centimeter  of  its  material  in  grams.  Strictly  considered, 
density  is  as  much  a  chemical  as  a  physical  property  of  matter, 
for  while  its  value  depends  partly  upon  the  closeness  with  which 
a  body's  molecules  are  packed  together,  i.e.,  the  number  of 
molecules  within  a  unit  volume,  it  is  affected  also  by  the  weights 
of  these  molecules — and  the  weights  of  the  molecules  depend 
upon  the  number,  the  nature,  and  the  arrangement  of  their 
component  atoms. 

Since  gravity  acts  upon  a  homogeneous  mass  of  matter  as  a 
whole  rather  than  upon  its  component  parts  independently,  the 
result  of  its  pull  cannot  be  governed  by  the  laws  of  symmetry. 
However,  in  the  case  of  dimorphous  substances  (substances 
crystallizing  in  two  different  systems)  the  two  forms  will  possess 
different  specific  weights.  Calcite  and  aragoni.e,  for  instance, 
are  both  forms  of  CaCO3,  but  the  specific  gravity  of  calcite, 
which  is  hexagonal,  is  2.714  and  that  of  aragonite,  which  is 
orthorhombic,  is  2 . 94. 

Methods  of  Determining  Densities. — Theoretically,  the 
simplest  method  of  determining  the  specific  gravity  of  a  solid 
body  is  (i)  to  weigh  a  small  fragment  suspended  from  one  arm  of 
a  balance  by  a  silken  thread,  (2)  to  immerse  in  water  and  weigh 
again,  (3)  to  subtract  the  weight  in  water  from  the  weight  in  air, 
and  (4)  to  divide  this  remainder  into  the  original  weight  in  air. 
The  reasons  for  these  manipulations  are  known  to  all  students 
familiar  with  the  principles  of  elementary  physics. 

If  a  fragment  whose  weight  in  air  is  3.25  grams  loses  1.25 
grams  when  weighed  in  water,  its  density  is  3.25-1-1.25  or  2.6. 

The  Jolly  Balance. — The  most  convenient  instrument  for 
the  rapid  determination  of  approximate  densities  is  the  Jolly 


MECHANICAL   PROPERTIES    OF    CRYSTALS 


193 


balance.     This  consists  essentially  of  a  spiral  of  wire  fastened  at 

the  top  to  a  movable  arm  (see  Fig.  289).     At  the  lower  end  it  is 

provided  with  two  little  pans,  one  suspended  beneath  the  other. 

The  lower  pan  is  kept  always  immersed  to  the  same  depth  in 

water,  while  the  other  one  hangs  in  the  air.     On  the  upright 

standard  behind  the  spiral  is  a  mirror  on 

which   is  engraved  or  painted  a  scale    of 

equal  parts.     The  object  whose  density  is 

to   be  determined  is  first  weighed  in  the 

upper  pan,  then  transferred  to   the  lower 

pan    and    weighed    again.      The     second 

weight  subtracted  from  the  first  will  give 

the   weight  of  the  water  replaced  by   the 

fragment.     The  original  weight  divided  by 

this  yields  the  result  desired. 

The  method  of  using  the  Jolly  balance  is 
simple.  Before  each  weighing  the  zero 
point  of  the  instrument  is  fixed  by  glancing 
along  the  end  of  the  wire  at  its  reflection  in 
the  mirror.  The  division  of  the  scale  that 
lies  exactly  in  the  line  of  sight  between  the 
point  on  the  spiral  and  its  image  in  the 
mirror  is  the  zero  point  for  that  determina- 
tion. A  small  fragment  of  the  substance 
whose  density  is  sought  is  then  placed  in  the 
upper  pan,  care  being  taken  to  prevent  the 
bottom  of  the  pan  from  touching  the  water 
in  which  the  lower  pan  is  immersed — 
usually  the  vessel  containing  the  water 
must  be  lowered  immediately  upon  placing 
the  fragment  in  the  upper  pan.  When  the 
instrument  comes  to  rest  and  the  vessel 
holding  the  water  is  adjusted  until  the  sur- 
face of  the  water  is  as  high  above  the  lower  pan  as  it  was 
originally,  a  second  reading  is  made.  The  fragment  is  then 
transferred  to  the  lower  pan,  and  the  level  of  the  water  is  again 
adjusted.  A  third  reading  is  now  taken.  If  the  first  reading 
13 


FIG.  289.— The  jolly 
balance  for  approximate 
determinations  of  specific 
gravity. 


194 


PHYSICAL   CRYSTALLOGRAPHY 


be  subtracted  from  the  second  and  third  readings  the  results  will 
give  the  relative  weights  of  the  fragment  in  air  and  in  water. 
The  difference  between  these  two  results  divided  into  the  first 
weight  will  yield  the  specific  gravity  sought. 

The  Use  of  Heavy  Solutions. — Often  it  is  more  convenient 
to  determine  the  density  of  small  fragments  of  crystals  by  some 
other  method  than  either  one  of  those  described  above.  This  is 
always  true  in  the  case  of  very  small  fragments  and  of  powders. 
For  the  determination  of  the  specific  gravity  of 
these  use  is  made  of  solutions  of  high  specific 
gravity,  which  by  the  addition  of  water  or  of 
some  other  liquid  may  have  their  densities 
lowered.  The  powder  or  small  fragment  is 
thrown  into  the  solution.  If  it  floats,  the 
density  of  the  solid  is  less  than  that  of  the 
liquid.  This  is  gradually  diluted  with  the 
proper  medium  until  the  fragments  remain 
suspended  in  the  mixed  fluids,  neither  rising 
nor  sinking.  When  this  occurs  fragments  and 
liquid  are  of  the  same  specific  gravity.  The 
density  of  the  liquid  may  then  be  determined 
by  any  one  of  the  physical  methods  appropriate 
for  this  purpose.  Naturally  the  use  of  heavy 
solutions  is  limited  to  those  substances  whose 
densities  are  less  than  their  own. 

There  are  serveral  solutions  in  common 
use  for  the  determination  of  specific  gravities.  The  one  most 
frequently  employed  is  a  solution  of  mercuric  and  potassium 
iodides,  known  as  the  Thoulet  solution.  Its  greatest  density  is 
3.19.  It  will  serve  to  determine  the  densities  of  substances 
lighter  than  3.19,  but,  of  course,  cannot  be  used  for  heavier 
substances.  Among  the  other  heavy  solutions  often  used  for  the 
same  purpose  may  be  mentioned  the  Klein  solution  (cadmium 
borotungstate  in  water)  with  a  maximum  density  of  3.6,  the 
Rphrbach  solution  (barium-mercury-iodide  in  water)  with  a 
density  of  3.59,  and  the  Retgers  solution  (thalium-mercuro- 
nitrate  at  76°)  with  a  specific  gravity  of  5.3. 


FIG.  290. — Sepa- 
rating funnel  for  use 
with  solutions. 


MECHANICAL  PROPERTIES  OF  CRYSTALS         195 

Practical  Uses   of  Specific   Gravity  Determinations.— 

Fragments  of  substances  that  closely  resemble  each  other  in  general 
appearance  and  in  many  of  their  physical  and  chemical  proper- 
ties may  often  be  easily  distinguished  from  one  another  by  their 
densities,  hence  the  determination  of  the  specific  gravity  of  a 
substance  under  investigation  will  often  lead  to  correct  inferences 
concerning  its  nature  and  identity.  For  example,  anorthite  and 
albite  are  two  feldspars  that  resemble  each  other  very  closely  in 
appearance.  The  former,  however,  has  a  specific  gravity  of 
2.76  and  the  latter  of  2 . 63. 

Further,  a  mixture  of  substances  in  a  finely  divided  form 
may  be  separated  into  its  component  parts  by  use  of  a  heavy 
solution  in  a  separating  funnel,  such  as  is  represented  in  figure 
290.  The  heaviest  of  the  powders  will  fall  and  can  be  drawn 
off  while  the  lighter  ones  remain  floating  on  the  solution. 

LIST    OF    SOME   IMPORTANT   MINERALS   AND    THEIR   DENSITIES. 

Sulphur    2.05  Topaz 3- 56 

Rock  salt 2.10  Garnet 3.75 

**    Gypsum 2 . 30  Celestite 3.97 

Orthoclase 2 .  56  Rutile 4.25 

Quartz 2 . 65  Magnetite 5 . 20 

Calcite    2.72  Hematite 5 . 30 

Muscovite    2.85  Cassiterite . 6 . 84 

Biotite 3.01  Cinnabar 8 .  oo 

METALS  AND    THEIR  APPROXIMATE   DENSITIES. 

Potassium 86  Iron    7 . 80 

Magnesium   ...    i .  80  Copper 8 . 90 

Aluminium 2.50  Silver   10.60 

Zinc 7 . 10  Gold 19 . 30 


CHAPTER  XVIII. 
OPTICAL  PROPERTIES  OF  CRYSTALS. 

Introduction. — The  light  that  falls  upon  a  crystal  is  partly 
reflected  from  its  surface,  partly  transmitted  through  its  sub- 
stance, and  partly  absorbed  within  it.  In  either  case  the  light 
suffers  changes  which  affect  its  character,  and  the  nature  of  these 
changes  is  determined  by  the  material  of  the  crystal  and  by  its 
physical  symmetry. 

Properties  Dependent  upon  Reflection. — Whenever  a  ray 
of  light  falls  upon  a  surface  a  portion  of  it  is  reflected  in  accordance 
with  the  simple  law :  the  angle  of  reflection  is  equal  to  the  angle  of 
incidence  and  both  are  in  the  same  plane.  Moreover,  some  of 
reflected  light  suffers  a  change  which  causes  it  to  possess  cer- 
tain properties  not  possessed  by  ordinary  light.  Because  of  the 
phenomena  exhibited  by  this  changed  part  of  the  reflected  light, 
this  part  is  described  as  being  polarized;  i.e.,  the  vibrations  that 
transmit  it  oscillate  in  a  single  plane  parallel  to  the  line  of  trans- 
mission. 

The  properties  depending  upon  the  character  of  the  reflected 
light  are  color  and  luster. 

Color. — The  color  of  a  crystallized  substance,  like  that  of 
other  substances,  depends  upon  the  character  of  the  light  reflected 
from  its  surface,  which  in  turn  depends  upon  the  color  of  the  light 
that  falls  upon  it,  upon  the  color  and  quantity  of  that  which  is 
absorbed,  and  upon  the  quality  of  the  reflecting  surface. 

When  illuminated  by  light  from  different  sources  the  same 
crystal  often  appears  quite  differently  colored.  The  most 
remarkable  case  of  this  kind  is  the  mineral  alexandrite,  or  chryso- 
beryl  (BeAl2O4),  the  value  of  which  as  a  gem  depends  upon  the 
fact  that  it  appears  green  by  daylight  and  red  by  lamplight  or 
gaslight.  Indeed,  the  colors  of  substances  vary  so  greatly 
in  different  lights  that  mineralogists  have  found  it  necessary  to 

196 


OPTICAL   PROPERTIES    OF   CRYSTALS  197 

agree  that  the  colors  mentioned  in  their  descriptions  of  minerals 
shall  refer  only  to  the  colors  in  daylight. 

When  wKite  light  falls  upon  any  surface  some  portion  of  it  is 
absorbed  or  transmitted,  the  remainder  being  reflected.  If 
certain  rays  are  absorbed  or  transmitted  to  a  greater  extent  than 
certain  others  the  reflected  light  will  be  of  a  different  color  from 
the  incident  light,  for  it  will  be  made  up  of  white  light  minus  the 
rays  that  have  been  absorbed  or  transmitted.  When  white  light 
falls  upon  a  mass  of  cinnabar  (HgS),  for  instance,  certain  of  its 
rays  are  absorbed.  Those  that  remain  and  are  reflected  consti- 
tute red  light,  and  the  mineral  appears  red. 

The  color  of  opaque  substances  is  much  more  characteristic 
than  that  of  transparent  or  translucent  ones.  In  the  latter  case 
the  distinctive  color  (ideochromatic)  is  often  obscured  by  im- 
purities or  by  some  pigment  (allochromatic)  which  is  present  in 
such  small  quantity  that  its  nature  in  many  instances  cannot  be 
determined.  Hence,  not  much  reliance  can  be  placed  in  the 
colors  of  such  substances  as  distinctive  characteristics.  For 
instance,  many  minerals  that  are  colorless  when  pure  are  found 
in  nature  with  a  great  variety  of  tints.  Tourmaline  is  a  case  in 
point.  While  transparent  colorless  crystals  are  known,  the 
predominant  colors  are  black,  brown,  green,  and  red. 

Although  opaque  substances  possess  much  more  character- 
istic colors  than  do  transparent  or  translucent  ones,  it  frequently 
happens  that  their  true  colors  are  obscured  by  a  surface  tarnish. 
The  true  color  is  the  color  on  a  fresh  fractured  surface. 

Streak. — Not  only  does  the  color  of  a  substance  vary  with  its 
nature,  but  it  varies  also  with  the  character  of  its  'reflecting 
surfaces.  More  light  is  reflected  from  a  smooth  surface  than 
from  a  rough  one,  and,  conversely,  more  light  is  absorbed  by  a 
rough  surface  than  by  a  smooth  one.  As  a  consequence,  the  light 
reflected  from  a  smooth  surface  and  that  from  a  rough  one  may  be 
made  up  of  different  components  differently  combined,  and  hence 
the  colors  of  the  two  surfaces  may  appear  different.  A  pyrite 
(FeS2)  crystal  has  a  bright,  brassy-yellow  color,  while  a  rough- 
surfaced  fragment  of  the  same  mineral  may  appear  bronzy- 
yellow  or  greenish. 


198  PHYSICAL   CRYSTALLOGRAPHY 

A  still  greater  contrast  in  color  is  often  noted  when  crystals  of  a 
substance  are  compared  with  their  powders.  This  is  especially 
true  of  opaque  minerals  and  of  translucent  ones  that  are  highly 
colored.  While  pyrite  crystals  are  brassy-yellow,  their  powder  is 
greenish-black.  Hematite  in  crystals  is  often  of  a  steel-black 
color.  Its  powder  is  blood-red  or  reddish-brown. 

The  color -of  the  powder  of  a  substance  is  more  characteristic 
than  that  of  its  crystals,  partly  because  in  the  powder  the  effect 
of  variations  in  the  quality  of  the  reflecting  surfaces  is  eliminated. 

The  most  convenient  method  of  obtaining  a  small  quantity 
of  crystal  powder  is  by  drawing  a  fragment  of  the  crystal  across  a 
rough  porcelain  surface.  The  mark  left  on  the  porcelain  is  known 
as  the  streak.  In  all  descriptions  of  mineral  species  the  color  of 
the  streak  is  mentioned. 

Transparent  and  translucent  substances,  whatever  their  color, 
usually  possess  a  white  streak.  This  is  not  so  much  due  to 
differences  in  the  absorbent  power  of  large  fragments  and  of 
their  powder  as  it  is  to  the  dilution  of  the  pigment  that  colors 
them.  On  the  same  principle  a  tumbler  of  water  to  which  a 
little  blue  litmus  solution  is  added  has  a  bluish  tint,  while  a 
single  drop  of  the  same  water  is  practically  colorless. 

Luster. — The  surfaces  of  crystals  often  present  very  decided 
peculiarities  independently  of  color.  Some  surfaces  reflect 
nearly  all  the  light  that  falls  upon  them,  others  but  a  small 
quantity.  The  former  glisten  perceptibly,  the  latter  are  dull. 
Others  reflect  a  portion  of  the  light  from  their  outer  surfaces,  and 
a  portion  from  the  surfaces  of  cracks,  etc.,  within  them.  Some 
others  scatter  the  light — diffuse  it.  The  result  of  these  different 
phenomena  is  known  as  luster. 

The  terms  most  frequently  used  in  describing  lusters  are  the 
following:  metallic,  vitreous,  adamantine,  resinous,  pearly,  and 
silky.  The  metallic  luster  is  confined  to  opaque  substances.  It 
is  the  luster  that  is  characteristic  of  the  metals.  The  vitreous 
luster  is  that  of  glass;  the  adamantine,  that  of  the  diamond  and  of 
other  very  brilliant  transparent  substances;  and  the  resinous,  that 
of  rosin.  The  pearly  luster  is.  found  only  in  those  substances  that 
have  a  very  perfect  cleavage  or  that  are  traversed  by  numerous 


OPTICAL    PROPERTIES    OF    CRYSTALS  199 

cracks.  The  play  of  colors  that  is  its  characteristic  feature  is 
caused  by  the  interference  of  a  portion  of  the  light  reflected  from 
the  sides  of  the  cleavage  or  other  cracks,  a  phenomenon  analogous 
to  that  of  Newton's  rings.  It  is  so  called  because  it  is  the  luster  of 
satin  and  of  silk. 

The  Transmission  of  Light. — That  portion  of  incident  light 
which  is  not  reflected  from  the  surface  of  a  substance  penetrates 
its  mass,  and  either  passes  through  it  or  is  stopped  by  it.  That 
portion  which  passes  through  is  said  to  be  transmitted.  The  por- 
tion that  is  stopped  is  said  to  be  absorbed.  Nearly  all  substances 
transmit  some  light,  though  often  to  such  a  slight  degree  that  they 
are  practically  opaque.  An  opaque  substance  transmits  no  light, 
therefore  it  appears  dark  when  viewed  from  the  side  opposite  to 
that  on  which  the  light  is  incident.  A  transparent  substance 
transmits  nearly  all  the  light  that  penetrates  its  mass;  i.  e.,  nearly 
all  the  light  that  is  not  reflected  from  its  surface.  When  viewed 
from  the  side  opposite  to  that  on  which  light  is  falling,  it  may  be 
white  or  it  may  possess  a  distinct  tint.  An  object  can  be  seen 
distinctly  through  a  transparent  substance.  A  translucent  sub- 
stance transmits  some  light,  but  not  enough  to  render  distinctly 
visible  an  object  viewed  through  it.  A  piece  of  iron  is  opaque, 
colorless  glass  is  transparent,  and  porcelain  translucent. 

Absorption  of  Light. — The  difference  between  opaque, 
transparent,  and  translucent  substances  is  due  to  the  difference 
in  the  quantity  of  light  absorbed  by  them.  The  opaque  bodies 
absorb  all  the  light  that  is  not  reflected,  the  translucent  ones 
absorb  a  considerable  quantity  of  it,  and  the  transparent  bodies 
almost  none. 

In  many  translucent  and  transparent  bodies  light  of  a  certain 
color  may  be  absorbed  while  that  of  a  different  color  is  transmitted. 
As  a  result  of  this  phenomenon  the  substance  may  have  a  distinct 
color  in  transmitted  light  which  is  usually  different  from  that 
produced  by  reflected  light.  Thin  sheets  of  gold  are  yellow  by 
reflected  light  and  green  by  transmitted  light.  The  color  of  the 
light  transmitted  may  be  determined  by  the  nature  of  the  sub- 
stance through  which  it  passes  or  it  may  be  determined  by  inclu- 
sions or  by  a  pigment  diffused  through  its  mass.  The  former  is 


200 


PHYSICAL   CRYSTALLOGRAPHY 


an  ideochromatic  color,  and  the  latter  allochromatic.  More- 
over, different  portions  of  crystals  may  transmit  light  of  different 
colors.  This  will  happen  when,  during  the  crystal's  growth,  the 
solution  from  which  it  separated  changed  in  composition  from 
time  to  time,  and  consequently  material  of  a  slightly  different 
character  was  successively  added  to  the  nucleus  already  in 
existence.  The  result  is  a  distribution  of  color  which  is  fre- 
quently zonal  (see  p.  150),  as  in  quartz,  tourmaline,  augite  or 


FIG.  291. — Thin  section  of  rock  show-  FIG.  292.— Vertical  section  of  ottrelile 

ing    zonal    structure     in     hornblende.  in  rock,  showing  regular  distribution  of 

Section  cut  parallel  to  the  basal  plane  color.     Magnified  50  diameters.   (After 

of      the      hornblende.     Magnified     50  Rosenbusch}. 
diameters.     (After  Rosenbusch). 

hornblende  (Fig.  291),  and  sometimes  disposed  in  accordance 
with  the  presence  of  certain  crystal  planes,  as  in  ottrelile  or  augite 
(Fig.  292). 

Not  only  do  different  substances  vary  in  their  powers  of  absorp- 
tion, but  the  same  substance  when  in  crystals  may  possess  this 
power  in  different  degrees  along  different  directions.  Light 
transmitted  in  a  certain  direction  through  a  crystal  may  thus  lose 
some  of  its  rays,  while  other  rays  may  be  lost  when  transmission 
occurs  through  some  other  direction.  Thus  a  crystal  may  appear 
differently  colored  when  viewed  by  transmitted  light  in  different 
directions. 

Pleochroism. — This  is  the  general  term  applied  to  the  prop- 
erty of  exhibiting  different  colors  in  different  directions.  If  a 
crystal  possesses  different  colors  when  viewed  in  two  different 


OPTICAL    PROPERTIES    OF    CRYSTALS  2OI 

directions,  it  is  said  to  be  dichroic;  if  it  exhibits  three  distinct 
colors  when  viewed  in  three  different  directions,  it  is  trichroic. 

In  the  case  of  amorphous  and  isometric  substances  the  same 
colored  light  is  absorbed  irrespective  of  the  direction  of  the  rays 
transmitted  through  them.  These  bodies  thus  exhibit  no  pleo- 
chroism.  In  crystals  possessing  a  principal  plane  of  symmetry 
the  absorption  in  the  direction  of  the  c  axis  is  often  different  from 
that  which  takes  place  in  the  direction  of  the  lateral  axes.  Hence 
these  crystals  may  be  dichroic. 

A  cube  of  dark  tourmaline,  an  hexagonal  mineral,  when  viewed 
through  the  direction  of  the  c  axis  appears  much  darker  than  when 
viewed  in  the  direction  at  right  angles  to  this,  and  may  possess 
different  colors  in  the  two  directions.  The  absorption  is  greater 
for  the  ray  transmitted  parallel  to  c  than  for  the  rays  transmitted 
parallel  to  the  plane  of  the  axes  a.  This  fact  is  represented  by  the 
formula  O  >  E.  The  symbol  O  refers  to  the  ray  transmitted  in  the 
direction  of  the  c  and  E  to  the  ray  transmitted  parallel  to  the 
principal  plane  of  symmetry  (see  also  page  205). 

Only  crystals  possessing  no  principal  planes  of  symmetry  may 
be  trichroic,  the  absorption  being  different  along  three  directions 
perpendicular  to  one  another;  consequently,  these  crystals  may 
appear  of  three  different  tints  when  viewed  along  these  three 
different  directions,  and  at  the  same  time  they  may  possess 
different  degrees  of  transparency  in  these  same  directions. 

The  mineral  cordierite,  an  orthorhombic  mineral,  is  often 
strongly  trichroic  in  light  blue,  dark  blue  and  yellowish-white 
tints.  Glaucophane,  a  monoclinic  mineral,  is  blue,  violet,  and 
yellowish-green.  Axinite,  a  triclinic  mineral,  is  colorless,  yellow, 
and  violet.  Nearly  all  transparent  minerals  are  pleochroic  in 
some  degree,  but  frequently  their  differences  in  tint  along  different 
directions  are  so  slight  that  they  can  be  detected  only  by  the  aid 
of  an  instrument  known  as  the  dichroiscope,  which  consists  simply 
of  a  small  rhomb  of  iceland  spar  (calcite)  mounted  in  a  brass 
tube  closed  at  both  ends  except  for  two  small  holes  which  serve 
as  peep-holes,  and  through  which  the  crystal  is  viewed.  The 
little  rhomb  causes  double  refraction  (see  pp.  204-205)  and  thus 
separates  the  two  rays. 


202  PHYSICAL   CRYSTALLOGRAPHY 

Relation  Between  Pleochroism  and  Crystal  Symmetry.— 
Only  crystallized  substances  can  exhibit  pleochroism,  and  not  all 
these  exhibit  it.  Opaque  crystals  can,  of  course,  not  exhibit  it. 
Moreover,  crystals  that  are  completely  transparent  in  every 
direction  cannot  exhibit  it.  These  are  colorless  in  all  directions 
because  there  is  no  absorption.  Isometric  crystals  likewise 
absorb  equally  in  all  directions  and  consequently  must  show  a 
single  color  by  transmitted  light.  Pleochroism  is  therefore 
limited  to  anisotropic  substances — substances  crystallizing  in 
any  system  but  the  isometric — because  only  in  anisotropic  sub- 
stances is  light  absorbed  differently  in  different  directions. 
Hexagonal  and  tetragonal  crystals  may  show  dichroism,  but  they 
cannot  exhibit  trichroism.  Orthorhombic ,  monoclinic,  and  triclinic 
crystals  may  be  trichroic* 

Fluorescence  and  Phosphorescence. — Some  substances  such 
as  fluorite  (CaF2)  possess  the  property  of  transmitting  light  of  a 
certain  color  while  at  the  same  time  radiating  light  of  some  other 
color.  This  property  is  known  as  fluorescence.  Fluorite  may 
appear  green  by  transmitted  light,  while  at  the  same  time  it 
glows  with  a  pale  blue  light.  This  property  of  fluorescing 
(from  the  name  of  the  mineral  best  exhibiting  it)  is  now  known 
to  be  possessed  by  many  crystals,  such  as  those  of  the  uranium 
minerals,  and  those  containing  fluorine  and  boron,  and  the  manu- 
factured substances  fuchsin  (C20H19N3.HC1),  fluorescein  (C20H12- 
O5),  and  magnesium  cyanplatinite  (MgPt(CN)4+  7  H2O). 

In  the  case  of  crystals  of  magnesium  cyanplatinite  and  of 
some  other  substances  it  is  noted  that  the  fluorescence  is  different 
in  color  from  different  faces.  In  the  cyanplatinite,  which  is 
tetragonal,  the  prismatic  faces  fluoresce  green  and  the  basal  planes 
bluish-red.  In  general,  similar  planes  fluoresce  alike,  dissimilar 
planes  may  fluoresce  differently. 

Other  substances,  for  example,  phosphorus,  possess  the  power 
of  giving  off  light  rays  in  the  dark  at  comparatively  low  temper - 

*  The  mineral  dealers  furnish  at  low  prices  little  plates  of  mica,  beryl,  cor- 
dierite,  tourmaline,  and  other  minerals  so  mounted  in  cork  cells  that  they  may 
readily  be  revolved  about  an  axis.  On  holding  the  plates  opposite  a  window  and 
revolving  them,  their  differences  in  color  along  different  directions  are  conveniently 
studied. 


OPTICAL   PROPERTIES    OF    CRYSTALS 


203 


atures.  Many  specimens  of  diamond  (C),  calcite  (CaCO3),  etc., 
glow  when  removed  from  the  presence  of  sunlight  into  a  dark 
room.  This  property  is  known  as  phosphorescence.  Other  * 
minerals,  such  asfluorite  (CaF2)  and  apatite  (Ca5P3O12Cl),  glow 
brightly  when  heated  to  a  temperature  far  below  that  of  red  heat. 
Others  become  luminous  when  their  crystals  are  rubbed  together, 
as,  for  instance,  sugar  and  zinc-blende  (ZnS),  and  others  glow 
when  subjected  to  the  action  of  the 
cathode  rays  or  of  radium  emana- 
tions. The  difference  in  the  be- 
havior of  natural  and  imitation 
gems  under  the  action  of  the  cathode 
rays  is  sometimes  made  use  of  in 
detecting  frauds. 

Refracted  Light. — If  a  ray  of 
light  traveling  through  air  fall  upon 
a  transparent  solid  or  liquid  body 
obliquely,  it  suffers  a  change  in 

direction  in  its  passage  through  the  body;  i.e.,  it  is  refracted. 
Both  the  incident  and  the  refracted  rays  are  in  the  same 
plane,  but  the  latter  is  bent  toward  the  perpendicular  to  the 
surface  of  the  body  at  the  point  of  incidence  of  the  former.  For 
any  given  substance  the  amount  of  refraction  or  bending  varies 
with  the  obliqueness  of  the  incident  ray,  the  refraction  being 
greater  as  the  angle  of  incidence  becomes  larger. 

If  in  figure  293  CB  represent  the  path  of  a  ray  of  light  falling 
on  the  surface  of  a  substance  at  B,  its  angle  of  incidence  is  C  B  A 
or  i.  This  ray,  upon  entering  the  substance,  suffers  a  change  in 
direction,  its  new  path  being  represented  by  B  R.  Its  angle  of 
refraction  is  r. 

The  Index  of  Refraction. — For  every  uncrystallized  substance 
and  every  substance  crystallizing  in  the  isometric  system  there  is 
always  a  definite  relation  existing  between  the  angle  of  incidence  and 
the  angle  of  refraction.  This  relation  is  a  constant  one  for  light 
of  a  given  color,  provided  the  medium  surrounding  the  body  is 
the  same  in  all  cases.  For  example,  the  relation  existing  between 
the  angle  of  incidence  of  a  ray  of  light  passing  through  air  and 


204  PHYSICAL   CRYSTALLOGRAPHY 

striking  a  surface  of  glass  obliquely  and  its  angle  of  refraction  in 
the  glass  is  always  the  same  no  matter  what  may  be  the  size  of  the 
angle  of  incidence.  The  ratio  is  expressed  by  the  formula 

n  = ,  in  which  n  is  the  constant,  i  the  angle  of  incidence, 

sin  r 

and  r  the  angle  of  refraction  (see  Fig.  293). 

The  greater  the  amount  of  the  bending,  or  refraction,  of  a  ray, 
the  smaller  the  angle  r,  hence  the  less  the  value  sin  r  as  compared 
with  the  value  sin  i,  and  the  greater  the  constant  n,  or  the  greater 
the  power  of  refraction  of  the  body  to  which  it  refers. 

The  ratio  n,  as  has  already  been  stated,  is  a  constant  for  all 
rays  of  light  of  the  same  color  entering  a  homogeneous,  amor- 
phous, or  isometric  body  from  a  given  medium,  as,  for  instance, 
air,  irrespective  of  their  obliquity.  It  differs,  however,  for 
different  substances,  and  varies  with  the  medium  traversed  by 
the  rays  before  they  enter  the  refracting  substance,  and  also  with 
the  color  of  the  light. 

It  is  because  of  the  differences  in  amount  of  refraction  suffered 
by  rays  of  different  colors  that  the  spectrum  is  produced  when 
white  light  is  allowed  to  traverse  a  glass  prism. 

When  the  constant  n  refers  to  the  refraction  of  a  ray  of  light 
of  a  definite  color  passing  from  a  vacuum  into  a  body,  it  is  known 
as  the  index  of  refraction.  It  is  characteristic  for  different  sub- 
stances, and  is  often  employed  for  identifying  them.  The  indices 
of  refraction  for  several  substances  are  as  follows: 

Water  =1.33.  Air  =1.0003.  Rock  salt  =  i.  54.  Diamond  =  2. 4195. 
Crown  Glass  =1.53.  Flint  glass  =1.61. 

Single  and  Double  Refraction. — In  all  amorphous  trans- 
parent bodies  and  in  all  those  substances  that  crystallize  in  the 
isometric  system  the  index  of  refraction  remains  the  same,  what- 
ever the  direction  of  the  incident  ray,  provided  the  light  is  the 
same  in  color  and  the  medium  through  which  it  passes  is  the 
same.  Such  bodies  are  known  as  isotropic  or  singly  refracting. 
In  them  there  is  only  one  refracted  ray. 

Light  falling  upon  substances  that  crystallize  in  any  other  system 
than  the  isometric  is  usually  split  up  by  refraction  into  two  rays 


OPTICAL   PROPERTIES    OF   CRYSTALS  205 

following  different  paths  (Fig.  294).  These  bodies  are  known  as 
aniso tropic  or  doubly  refracting.  Both  of  the  refracted  rays  are 
polarized. 

An  excellent  example  of  a  doubly  refracting  substance  is  a 
cleavage  piece  of  calcite  or  iceland  spar  (CaCO3),  which  on  ac- 
count of  its  strong  doubly  refracting  power  has  long  been  called 
" double-spar."  If  a  plate  of  the  clear  mineral  be  placed  over  a 
pin-hole  in  a  piece  of  cardboard  and  both  be  held  up  to  the  light, 
the  pin-hole  will  appear  double  (Fig.  294).  If  the  plate  be  now 
revolved  parallel  to  the  plane  of  the  cardboard  one  of  the  pin- 


FiG.  294. — Photograph  of  double  image  of  pin-hole  as  seen  through  a  cleavage 
rhombohedron  of  calcite. 

hole  images  will  revolve  about  the  other,  which  will  remain  fixed 
in  position.  The  light  ray  that  produces  the  fixed  image  may 
easily  be  shown  to  obey  the  law  of  simple  refraction — its  index  of 
refraction  is  a  constant.  This  is  called  the  ordinary  ray.  The  ray 
that  produces  the  other  image  obeys  another  law.  It  is  called  the 
extraordinary  ray.  Its  index  of  refraction  varies  with  the  direc- 
tion pursued  by  the  transmitted  ray.  In  calcite  the  index  of 
refraction  of  the  ordinary  ray  (written  o>)  is  i .  6585  for  yellow 
light,  while  that  of  the  extraordinary  ray  varies  between  i .  6585  and 
1.4863.  The  value  that  differs  most  from  to  is  taken  as  the 


206 


PHYSICAL   CRYSTALLOGRAPHY 


index  of  refraction  for  the  extraordinary  ray  (written  e).  In 
this  instance  w  >  e.  The  mineral  is  said  to  be  optically  negative. 
In  quartz,  on  the  other  hand,  <o  for  yellow  light  is  i .  5442  and  g 
is  i .  5533;  the  mineral  is  positive. 

The  explanation  of  the  double  image  seen  in  calcite  is  readily 
understood  by  reference  to  figure  295.  The  ray  BC  striking  the 
calcite  plate  at  C  is  transmitted  through  it  along  two  paths,  CD 
and  CE.  These  emerge  as  two  rays,  DF  and  EG.  The  eye 
receiving  the  light  at  F  sees  the  ray  as  though  originating  at  I, 
while  the  light  received  along  the  line  EG  appears  to  oriinate  gat 
J.  Hence,  two  images  of  the  source  of  light  are  seen — one  at  I 


FIG.  295. 

and  the  other  at  J.     If  a  black  spot  be  at  B,  likewise  two  images 
of  the  spot  will  be  seen  at  I  and  J. 

.  Uniaxial  and  Biaxial  Crystals. — In  some  anisotropic 
bodies,  like  quartz  and  calcite,  one  of  the  two  rays  into  which  the 
refracted  light  is  divided  possesses  a  constant  index  of  refraction, 
while  the  other  possesses  a  variable  index.  In  other  anisotropic 
bodies  both  refracted  rays  possess  variable  indices.  Neither  can 
be  spoken  of  as  the  ordinary  ray;  both  are  extraordinary.  The 
first  class  of  bodies  embraces  substances  that  crystallize  in  the 
hexagonal  and  tetragonal  systems;  i.e.,  in  systems  possessing  a 
principal  crystallographic  axis  (the  vertical  axis)  differing  from  a 
series  of  equivalent  lateral  axes.  The  second  class  embraces 
orthorhombic,monoclinic  and  triclinic  crystals,  or  those  with  three 
unequivalent  crystallographic  axes. 


OPTICAL   PROPERTIES    OF   CRYSTALS 


207 


In  bodies  of  the  first  class  there  is  one  direction;  viz.,  parallel  to 
the  vertical  axis,  along  which  a  ray  is  transmitted  without  double 
refraction.  This  is  the  ordinary  ray.  In  bodies  of  the  second 
class  there  are  two  directions  along  which  no  double  refraction  takes 
place,  and  these  two  directions  are  inclined  to  one  another  at  angles 
varying  with  the  body  and  with  the  color  of  the  light. 

Optical  Axes  and  the  Axial  Angles. — The  directions  in 
anisotropic  bodies  along  which  no  double  refraction  takes  place 
are  known  as  optical  axes.  Crystals  of  the  first  class,  as  defined 
above,  possess  but  one  of  these  axes,  hence  they  are  said  to  be 
uniaxial.  Those  of  the  second  class  possess  two  optical  axes, 
and  are  consequently  said  to  be  biaxial. 

The  angle  of  inclination  between  the  optical  axes  of  biaxial 


A' A 


A  A 


o  o, 


FIG.  296. 


FIG.  297. 


crystals  is  known  as  the  optical  axial  angle.  It  can  very  readily 
be  measured.  When  once  known,  it  serves  as  an  important  means 
for  distinguishing  between  two  substances  resembling  each  other 
in  other  respects. 

In  figure  296  is  represented  a  section  through  a  biaxial  crystal 
in  the  plane  of  the  optical  axes.  The  light  is  supposed  to  pass 
into  the  lower  side  of  the  crystal  as  a  bundle  of  rays  converging 
within  the  crystal.  The  lines  AO  represent  the  directions  along 
which  there  is  no  double  refraction  of  red  rays,  and  A'O',  the 
directions  along  which  there  is  no  double  refraction  of  violet 
rays.  The  optical  angle  for  the  red  ray  is  measured  by  the  arc 
between  A  and  A  and  that  for  violet  light  by  the  arc  between  A' 
and  A'. 


208  PHYSICAL   CRYSTALLOGRAPHY 

The  value  of  the  optical  angle  as  given  in  the  text-books  en 
crystallography  is  usually  the  true  axial  angle;  i.e.,  the  axial 
angle  within  the  crystal.  This  may  be  determined  from  the 
apparent  axial  angle,  the  value  of  which  is  obtained  by  observation 
in  air.  The  apparent  angle  is  always  greater  than  the  true  angle 
(Fig.  297),  for  the  rays  of  light,  passing  from  the  denser  crystal 
into  the  rarer  medium  surrounding  it,  are  bent  away  from  the 
perpendicular,  and  so  are  inclined  to  each  other  at  a  greater  angle 
than  while  traversing  the  denser  crystal.  In  the  figure,  b  is  the 
apparent  optical  axial  angle  and  a  the  true  angle.  In  order  to 
reduce  the  size  of  the  apparent  axial  angle  so  that  it  may  be  seen 
more  easily  the  observation  is  frequently  made  upon  a  crystal 
plate  immersed  in  water,  oil,  or  in  some  highly  refracting  liquid. 
From  the  data  obtained  for  this  observation  the  true  axial  angle 
is  calculated. 

The  apparent  axial  angles  of  hypersihene,  bronzi'e,  and  en- 
siatite  when  measured  in  oil  are  85°  39',  112°  30',  and  133°  8'. 
For  olivine  (Mg2SiO4)  the  apparent  and  true  axial  angles  for 
differently  colored  rays  are: 


I 
Red                    Yellow 

Blue 

Apparent  angle  in  oil    . 

100°    <^2'                        101°   2' 

101°      0' 

True  angle  

86°  i'                    86°  10' 

86°  32' 

NOTE. — The  relations  of  crystallized  minerals  toward  light 
rays  transmitted  through  them  are  too  complicated  to  be  discussed 
at  greater  length  in  an  elementary  text-book  of  this  kind.  Suffice 
it  to  state  that  the  simple  principles  enunciated  above  have  been 
made  the  basis  of  much  study,  and  that  the  discussions  to  which 
this  study  has  given  rise  are  treated  in  large  volumes  devoted 
exclusively  to  the  optical  properties  of  minerals.  By  means 
of  these  properties  the  physical  symmetry  of  crystals  is  most 
beautifully  shown.  To  one  who  intends  making  a  special  study 
of  crystals  or  a  study  of  rocks,  a  knowledge  of  the  optical 
properties  of  crystallized  substances  is  indispensable. 


CHAPTER  XIX. 
THERMAL  PROPERTIES  OF   CRYSTALS. 

Transmission  of  Heat. — Since  heat  and  light  are  so  similar 
physically,  it  is  not  to  be  wondered  at  that  crystals  conduct 
themselves  toward  heat  rays  very  much  as  they  do  toward  rays 
of  light.  Just  as  there  are  substances  that  are  opaque  and  others 
that  are  transparent  to  light,  there  are  those  that  are  opaque  and 
those  that  are  transparent  to  heat.  Substances  that  are  trans- 
parent to  light  are  not  necessarily  transparent  to  heat,  and  vice 
versa.  Sylvite  (KC1),  rock  salt  (NaCl),  and  alum  (KA1(SO4)2  + 
12  H2O)  are  all  transparent  to  light.  Only  the  first  two,  however, 
will  allow  heat  to  pass  through  them.  Alum  is  practically 
opaque  to  it.  Substances  that  are  transparent  to  heat  are  said  to  be 
diathermous. 

Alum  is  opaque  to  heat  when  in  solution,  as  well  as  when  in 
crystals.  Tanks  of  alum  solution  are  often  placed  between  the 
source  of  the  light  in  projecting  lanterns  and  the  object  the  image 
of  which  is  desired,  in  order  to  cut  off  the  heat  and  protect  the 
object  from  injury. 

Most  of  the  laws  that  are  known  to  govern  the  action  of 
crystals  toward  transmitted  light  are  found  to  be  applicable  also 
in  the  case  of  heat.  For  instance,  there  are  crystals  that  are 
singly  refracting  and  others  that  are  doubly  refracting  toward 
heat  rays,  and  of  the  doubly  refracting  crystals  some  are  uniaxial 
and  others  biaxial. 

There  are  a  few  results  of  the  action  of  heat  upon  crystals, 
however,  that  find  no  analogies  in  the  action  of  light  upon  them; 
by  conduction  crystals  (together  with  all  other  substances) 
become  heated  to  their  most  distant  parts;  by  the  addition  of  a 
moderate  amount  of  heat  they  expand  or  contract,  and  by  the 
addition  of  a  greater  quantity  they  melt  or  become  vaporized. 

Conduction  of  Heat. — Substances  differ  in  their  capacity 
14  209 


210  PHYSICAL    CRYSTALLOGRAPHY 

to  conduct  heat.  Of  the  metals,  silver  is  the  best  conductor  and 
bismuth  the  worst.  Non-metallic  substances  are  poorer  con- 
ductors than  is  bismuth.  They  vary  greatly  in  conductive 
capacity,  just  as  do  the  metals.  Moreover,  crystallized  substances 
conduct  with  different  rates  in  different  directions. 

The  relative  conductive  capacities  of  a  crystal  in  different 
directions  may  be  investigated  by  covering  its  faces  with  a  thin 
film  of  wax  and  touching  the  wax  with  a  needle  point  kept  hot 
by  an  electric  current.  The  wax  will  melt  around  the  hot  point 

for  distances  varying  with  the  rapidity 
with  which  the  heat  is  conducted  in 
different  directions.  By  inspection  of  the 
melted  area  the  relative  conductive  ca- 
pacity in  these  different  directions  may 
easily  be  determined.  The  lines  drawn 
FIG.  298.— Diagrammatic  around  the  melted  area  at  different 

illustration  of   variation  in  ...  -,  11    j 

rate  of  heat  conductivity  in  stages  in  its  production  are  called 
biaxial  crystals.  isothermal  lines.  They  constitute  curves 

analogous  to  the  curves  of  hardness. 

The  isothermal  lines  on  the  surfaces  of  amorphous  bodies  and 
on  the  faces  of  isometric  crystals  are  always  circles,  indicating  that 
in  these  bodies  the  conductive  capacity  does  not  vary  with  direc- 
tion. In  the  case  of  hexagonal  and  tetragonal  crystals  (uniaxial 
crystals),  the  isotherms  on  basal  planes  are  circles,  but  those  on 
the  other  faces  are  ellipses.  These  curves  indicate  that  while 
conduction  is  equally  rapid  in  directions  at  right  angles  to  the 
vertical  axis,  in  other  directions  its  rate  is  either  greater  or  less 
than  this,  and  in  the  direction  of  the  vertical  axis  it  is  a  maximum 
or  minimum.  The  isotherms  on  the  faces  of  crystals  in  the 
remaining  systems  show  that  the  conductivity  varies,  being 
greatest  and  least  along  directions  perpendicular  to  one  another 
and  intermediate  along  intermediate  directions  (see  Fig.  298). 

Expansion  and  Contraction. — Solids  and  liquids,  as  a  rule, 
expand  when  heated  and  contract  when  cooled,  the  change  in 
volume  varying  with  the  substance. 

The  ratio  between  the  length  (1)  of  a  bar  of  a  given  substance 
at  o°  and  its  increase  in  length  when  heated  to  100°  (l'  =  length 


THERMAL   PROPERTIES    OF    CRYSTALS  211 

at  100°)  is  known  as  its  coefficient  of  expansion.  This  is 
a  constant  for  each  substance  when  measured  under  similar 
conditions. 

Exceptions  to  the  above  rule  exist,  but  they  are  confined  to 
substances  when  near  their  fusing  points;  i.e.,  when  they  are 
about  to  change  their  state  from  liquid  to  solid  or  vice  versa. 

When  subjected  to  a  varying  temperature,  amorphous  substances 
and  isometric  crystals  expand  and  contract  equally  in 
different  directions;  i.e.,  they  possess  but  a  single 
coefficient  of  expansion,  while  uniaxial  crystals 
expand  and  contract  more  or  less  in  the  direction  of 
their  vertical  axis  than  in  any  other  direction,  and 
equally  in  directions  perpendicular  to  this  axis ;  i.e., 
they  possess  a  maximum  and  a  minimum  coefficient 
of  expansion.  Orthorhombic ,  monoclinic  and  tri- 
clinic  crystals  (biaxial  crystals)  expand  and  contract  FIG.  299. 
differently  in  different  directions. 

Although  not  discernible  to  the  eye,  the  differences  in  length 
caused  by  varying  temperature  in  the  case  of  the  axes  of  all 
crystals  but  those  belonging  to  the  isometric  system  are  sufficiently 
great  to  affect  the  values  of  their  interfacial  angles  and,  conse- 
quently, their  axial  ratios  (see  Fig.  299).  Hence,  in  the  deter- 
mination of  axial  ratios,  it  is  often  necessary  to  note  the  temper- 
ature at  which  the  measurement  serving  as  the  basis  of  calculation 
was  made  in  order  that  the  ratio  may  have  a  definite  meaning. 

Crystals  of  quartz  (hexagonal  SiO2)  increase  by  .000781  of 
their  lengths  and  .001419  of  their  breadths  when  their  temper- 
ature is  increased  100°  C.  Their  a  axes  increase  in  length  about 
twice  as  much  as  their  c  axes,  and  thus  their  axial  ratio  dimin- 
ishes with  an  increase  in  temperature. 

Fusion  and  Vaporization. — When  the  temperature  of  a  body 
is  increased  sufficiently,  if  it  is  not  decomposed,  it  becomes 
vaporized,  usually  passing  through  the  liquid  state  before  becom- 
ing a  gas.  The  temperature  at  which  a  body  changes  from  a 
liquid  to  a  solid,  or  vice  versa,  is  known  as  its  fusing  point;  the 
temperature  at  which  it  changes  to  a  gas  is  its  boiling  point.  The 
fusing  and  boiling  points  differ  for  different  substances,  but  are 


212  PHYSICAL   CRYSTALLOGRAPHY 

constant  for  the  same  substances  under  similar  conditions.  Since 
they  are  determined  by  the  nature  of  the  molecules  of  which  the 
body  is  composed,  and  not  upon  their  arrangement,  they  cannot 
be  governed  by  the  laws  of  symmetry. 


TABLE    OF    FUSING   POINTS. 

Metals. 

Tin 228°  Silver 960° 

Cadmium 320°  Gold 1062° 

Lead 335°  Copper 1083° 

Zinc 419°  Nickel    1452° 

Antimony 630°  Cobalt    1490° 

Aluminium 658°  Platinum   ....  1755° 

Non-metallic  Substances. 

Ice o°         Albite   (NaAlSi3O8)  . . .    1230° 

Sulphur 114-5°      Fluorite   (CaF2)   1387° 

Nitre 345°          Anorthite  (CaAl2(SiO4)2)  1532° 

Scale  of  Fusibility. — The  exact  determination  of  the  fusing 
point  of  a  crystallized  body  often  serves  as  an  important  aid  in 
identifying  it.  The  methods  used  for  this  purpose  are,  however, 
either  very  complicated,  as  in  the  case  of  substances  with  high 
melting  points,  or,  when  simple,  they  are  especially  adapted  to 
the  determination  of  low  fusing  points,  such  as  those  of  many 
organic  compounds.  The  latter  are  described  in  manuals  on 
organic  chemistry,  and  the  former  in  treatises  on  heat  or  in  special 
articles  on  the  measurement  of  high  temperatures. 

Although  it  is  not  always  practicable  to  determine  the  accurate 
melting  point  of  a  substance,  it  is  frequently  important  to  obtain 
at  least  some  notion  of  its  relative  fusibility  as  compared  with 
other  substances.  For  this  purpose  the  mineralogist  von  Kobell 
has  proposed  the  following  seven  minerals  arranged  in  the  order 
of  their  fusibilities,  to  serve  as  a  scale  with  which  to  compare 
other  substances. 


THERMAL   PROPERTIES    OF    CRYSTALS 


2I3 


VON  KOBELL'S  SCALE  OF  FUSIBILITY. 

1.  Stibnite.  Melts  in  the  flame  of  a  candle. 

2.  Natrolite.         Readily  melts  to  a  globule  in  a  blowpipe 

flame. 

3.  Almandine.     Difficultly  fusible  to  a  globule  in  the  blowpipe 

flame. 

4.  Asbestus.         The  ends  of  fine  splinters  fuse  to  globules 

before  the  blowpipe. 

5.  Orthoclase.     The  sharp  corners  of  small  fragments  may 

be  rounded  before  the  blowpipe. 

6.  Bronzite.         The  sharp  corners  of  small  fragments  show 

the  merest  trace  of  rounding  in  the  flame  of 
powerful  blowpipe. 

7.  Quartz.         Infusible. 

Minerals  are  said  to  fuse  at  i,  2,  3,  etc.,  when  they  melt  as 
easily  as  stibnite,  natrolite,  almandine,  etc. 

Very  small  fragments  should  always  be  used  in  comparing 
fusibilities.  They  may  be  held  in  the  loop  of  a  piece  of  platinum 
wire  fused  into  the  end  of  a  glass  tube. 


CHAPTER  XX. 
ELECTRICAL  AND  MAGNETIC  PROPERTIES  OF  CRYSTALS. 

Magnetism  of  Bodies. — All  substances  are  believed  to  be 
magnetic  in  some  degree,  although  in  most  of  them  the  magnetic 
power  is  so  slight  as  to  be  overlooked. 

A  few  metals  like  iron  and  compounds  of  metals  like  magnetite 
(Fe3O4)  are  strongly  magnetic,  often  possessing  poles  like  an 
ordinary  magnet.  The  majority  of  compounds,  however,  are 
not  endowed  with  the  property  of  attracting  other  bodies,  though 
they  may  themselves  be  attracted  by  or  repelled  from  a  strong 
magnet.  Those  substances  that  are  attracted  by  both  poles  of  an 
ordinary  magnet  are  said  to  be  paramagnetic,  while  those  that  are 
repelled  from  both  poles  are  diamagnetic.  When  rods  of  a  para- 
magnetic substance  are  suspended  between  the  poles  of  a 
U-shaped  electromagnet,  they  place  themselves  with  their  long 
axes  in  the  line  joining  its  poles.  If  the  rods  be  of  diamagnetic 
substances,  their  long  axes  take  positions  at  right  angles  to  this 
line. 

The  force  with  which  anisotropic  crystals  are  attracted  or 
repelled  by  a  magnet  varies  with  the  direction  along  which  it  acts. 
The  directions  along  which  greatest  and  least  forces  are  exerted 
accord  with  the  directions  of  the  axes  of  morphological  symmetry 
just  as  do  the  directions  of  heat  propagation,  etc. 

Amorphous  bodies  and  isometric  crystals  are  similarly  para- 
magnetic or  diamagnetic  in  all  directions.  In  tetragonal  and 
hexagonal  crystals  the  magnetic  force  exerts  a  maximum  and  a 
minimum  power  in  directions  parallel  with,  and  perpendicular  to, 
the  vertical  axis,  and  in  all  directions  perpendicular  to  the  axis  it  is 
equal.  In  crystals  of  a  lower  grade  of  symmetry  the  force  varies 
in  three  rectangular  directions. 

Electrical  Properties. — Crystals  may  become  charged  with 
electricity  by  one  or  another  of  many  ways.  Many  may  be  made 

214 


ELECTRICAL  AND    MAGNETIC    PROPERTIES    OF    CRYSTALS     215 

electrical  by  rubbing,  others  by  compression,  others  by  fracturing, 
others  by  heating,  etc.  Only  those  crystals  that  are  bad  con- 
ductors manifest  electrical  properties  to  any  great  degree,  since 
in  the  case  of  good  conductors  the  electricity  flows  away  as  fast  as 
it  is  generated  unless  the  crystals  be  insulated.  Crystallized  sub- 
stances may  therefore  be  separated  into  two  groups:  (a)  con- 
ductors, which  may  be  made  electrical  but  which  do  not  retain 
their  electricity,  and  (b)  non-conductors,  which,  when  they  be- 
come electrified,  maintain  their  electrical  condition  for  some  time. 

Conduction  of  Electricity. — Crystals  that  are  good  electrical 
conductors  behave  toward  electricity  in  a  manner  analogous  to 
the  behavior  they  exhibit  with  respect  to  heat.  Amorphous  and 
isometric  minerals  conduct  with  equal  facility  in  different  directions. 
Those  with  one  principal  plane  of  symmetry  conduct  most  rapidly 
along  a  direction  parallel  to  the  c  axis,  and  least  rapidly  along 
directions  perpendicular  thereto,  or  vice  versa.  The  optically 
biaxial  minerals  conduct  with  different  degrees  of  facility  along 
different  directions. 

Thermoelectrical  Properties. — When  two  strips  of  different 
metals  are  brought  into  contact  at  both  ends  and  one  end  of  the 
" couple"  is  heated,  an  electrical  current  is  generated.  The 
metal  from  which  the  current  flows  at  the  cold  end  of  the  couple 
is  positively  thermoelectric,  and  the  other  negatively  thermoelec- 
tric. In  general,  when  any  two  conductors  are  so  united  and 
subjected  to  differences  in  temperature  at  their  different  junctions, 
a  current  of  electricity  will  flow  from  one  to  the  other. 

Many  crystallized  minerals  that  are  good  conductors  exhibit 
thermoelectrical  properties  to  a  marked  degree.  When  two 
crystals  of  pyrite  are  brought  into  contact  and  heated,  a  very 
strong  current  is  set  up  between  them,  provided  the  crystals  are 

not  morphologically  identical.     Pyritoids  (  -      - )  of  pyrite  whose 

faces  are  striated  parallel  to  the  cubic  faces  are  thermoelectrically 
positive  (Fig.  300,  while  those  striated  perpendicularly  to  these 
faces  are  negative  (Fig.  301).  The  current  thus  generated  is 
stronger  than  the  strongest  current  generated  in  a  similar  manner 
between  any  two  metals. 


2l6  PHYSICAL   CRYSTALLOGRAPHY 

The  force  of  the  current  generated  between  two  different 
crystal  individuals  varies  with  the  nature  of  these  individuals 
and  with  their  morphological  features.  The  best-known 
producers  of  thermoelectric  currents  are  hemihedral  and 
tetartohedral  crystals. 

Thermoelectrical  currents  are,  however,  not  limited  to 
" couples"  of  individual  crystals.  Rods  cut  from  a  single  crystal 
will  often  generate  a  thermoelectrical  current  when  they  are  so 
placed  in  contact  that  the  same  crystallographic  directions  in 


FIG.  300.  FIG.  301. 

Two  crystals  of  pyrite  illustrating  relation  between  striations  and  thermo- 
electrical properties  of  this  mineral.  Fig.  300  represents  a  positive,  and  Fig.  301 
a  negative  crystal. 

them  are  not  parallel  to  one  another.  A  law  of  direction  thus 
applies  to  the  exhibition  of  the  thermoelectrical  as  well  as  to  that 
of  the  other  properties  of  crystals. 

Pyroelectricity  and  Pyroelectrical  Properties. — Non-con- 
ductors of  electricity  when  crystallized  exhibit  certain  peculiar 
electrical  phenomena  different  from  those  exhibited  by  con- 
ductors. 

In  many  instances  when  the  temperature  of  a  crystal  or  of  a 
fragment  of  a  crystal  is  rapidly  changing  it  becomes  charged 
with  electricity,  one  end  becoming  positively  electrified  and 
other  end  negatively  charged.  That  end  charged  with  positive 
electricity  during  the  rising  of  the  temperature  and  with  negative 
electricity  when  the  temperature  is  falling  is  called  the  analogue 
pole.  The  end  which  is  negatively  charged  with  rising  temper- 
ature arid  positively  charged  with  falling  temperature  is  the 
antilogue  pole.  The  electricity  produced  in  a  body  by  changes  in 
temperature  is  known  as  pyroelectricity. 

The  distribution  of  pyroelectricity  is  governed  entirely  by  the 


ELECTRICAL  AND    MAGNETIC    PROPERTIES    OF    CRYSTALS     21 7 


symmetry  of  the  crystals  that  exhibit  it.  Hemimorphic  crystals 
have  the  two  ends  of  their  unsymmetrical  axis  differently  charged, 
as  have  also  all  other  crystals  possessing  a  single  polar  axis  of 
symmetry,  such  as  certain  hemihedral  and  tetartohedral  forms.  In 
these  cases  fragments  of  the  crystals  exhibit  the  same  pyro- 
electric  properties  as  complete  crystals. 

In  crystals  with  several  polar  axes  of  symmetry  the  electricity  is 
distributed  over  faces  and  solid  angles  in  such  a  manner  as  to  bring 
out  clearly  the  symmetry  of  the  form — morphologically  similar 


FIG.  302. — Tourmaline 
crystal  after  powdering 
with  a  mixture  of  sulphur 
and  minium.  Its  upper 
end  is  the  analogue  pole. 


FIG.  303.— Boracite  crystal 
dusted  with  sulphur  and 
minium  to  show  distribution 
of  pyroelectric  properties. 
The  darker  dotting  represents 
the  minium. 


parts  always  being  similarly  electrified.  In  general,  the  opposite 
ends  of  symmetry  axes  become  similarly  electrified  when  they  are 
equivalent  morphologically,  and  differently  electrified  when 
they  are  morphologically  unequivalent. 

Amorphous  substances  and  holohedral  regular  crystals  show 
no  pyroelectrical  phenomena. 

The  pyroelectrical  properties  of  crystals  may  be  conveniently 
investigated  with  the  aid  of  very  simple  apparatus.  The  crystal 
to  be  studied  is  placed  in  an  air-bath  and  brought  to  a  fairly  high 
temperature.  It  is  then  taken  from  the  air-bath,  quickly  passed 
through  the  flame  of  a  spirit  lamp,  and  then  laid  on  a  piece  of 
paper  in  a  cold  room.  After  it  has  lain  undisturbed  for  a  few 
moments  it  is  dusted  through  a  little  sieve  with  a  mixture  of 


2l8  PHYSICAL    CRYSTALLOGRAPHY 

pulverized  sulphur  and  minium.  The  sulphur  in  passing  through 
the  meshes  of  the  sieve  becomes  negatively  electrified  and 
the  minium  positively  charged.  The  former  is  attracted  to 
the  positive  poles  of  the  crystal  which  are  thus  colored  yellow, 
and  the  minium  to  the  negative  poles  which  are  colored  red. 
The  distribution  of  the  two  colors  and  their  intensity  cor- 
respond to  the  distribution  and  intensity  of  the  different  charges 
of  electricity  on  the  crystal. 

Figures  302  and  303  represent  the  appearance  of  a  cooling 
tourmaline  crystal  and  of  a  cooling  boracite  crystal  after  dusting 
with  sulphur  and  minium.  The  tourmaline  is  hemimorphic.  Its 
upper  end  is  negatively  electrified — it  is  the  analogue  pole — and 
its  lower  end  positively  electrified.  This  is  the  antilogue  pole. 
The  boracite  crystal  is  apparently  a  combination  of  inclined  hemi- 
hedral  forms  of  the  isometric  system.  The  symmetry  of  its  pyro- 
electric  properties  is  in  accord  with  the  morphological  symmetry 
of  such  a  combination. 


CHAPTER  XXI. 
SOLUTION  AND  ETCHED  FIGURES. 

When  a  substance  is  subjected  to  the  action  of  solvents 
its  corners  will  usually  be  dissolved  more  rapidly  than  the 
plane  or  curved  surfaces  upon  it,  and  no  definite  laws  of  solu- 
tion will  be  observed  unless  the  substance  be  a  crystal  or  a  portion 
of  a  crystal.  The  rapidity  with  which  the  different  parts  dissolve 
is  governed  by  the  conditions  under  which  the  body  exists  with 
respect  to  the  solvent.  In  the  case  of  crystallized  bodies,  however, 
the  rapidity  of  solution  is  governed  largely  by  the  laws  of  sym- 
metry. In  every  crystal  there  are  directions  along  which  solu- 
tion occurs  at  different  rates.  Under  similar  conditions  "  the 
crystals  of  a  given  body  will  always  dissolve  more  rapidly  along  a 
certain  direction  than  along  any  other  direction.  Under  different 
conditions,  however,  the  maximum  and  minimum  directions  may 
be  different.  Indeed,  a  maximum  direction  for  one  solvent  may 
become  a  direction  of  minimum  solution  for  some  other  solvent. 
A  similar  condition  is  true  with  respect  also  to  decomposing  agents. 
A  crystal  will  often  suffer  decomposition  at  different  rates  along 
different  directions. 

A  crystal  of  the  isometric  mineral  ftourite  (CaF2)  when 
subjected  for  a  minute  to  the  action  of  a  dilute  solution  of  hydro- 
chloric acid  (one  part  of  a  36  per  cent,  solution  of  HC1  in  i  part 
of  H2O)  loses  a  layer  .006  mm.  thick  from  the  ooOoo  faces,  .01 
mm.  from  the  O  faces,  and  a  little  more  from  the  oo  O  faces. 
When  treated  with  a  concentrated  solution  of  soda  for  the  same 
length  of  time  the  oo  O  oo  faces  lose  a  layer  of  material  .  016  mm.  in 
thickness,  while  the  O  faces  lose  only  .  006  mm.  and  the  oo  O  fac^s 
.007  mm. 

At  the  same  time  that  the  solvent  is  dissolving  the  exposed 
portions  of  a  crystal  it  may  eat  its  way  into  its  interior  along  planes 
that  may  or  may  not  be  identical  with  the  cleavage  planes.  These 

219 


220 


PHYSICAL   CRYSTALLOGRAPHY    ' 


FIG.  304. — Inclusions 
produced  along  solution 
planes  in  augite  by  the 
alteration  of  its  s  u  In- 
stance. Magnified. 


planes  are  known  as  solution  planes.     Their  positions  are  deter 
mined  by  the  nature  of  the  solvent,  that  of  the  substance  of  the 
crystal  and  its  symmetry.     Often  the  products  formed  by  the 
solution  are  precipitated  in  the  places  where 
formed,  or  the  cavities  produced  by  their 
removal  are  subsequently  filled  with  some 
new  substance  and  thus  inclusions  result. 
Inclusions  of  this  kind  often  indicate  their 
mode    of   origin   by   their  arrangement  in 
definite  planes. 

The  accompanying  figure  (Fig.  304) 
represents  the  appearance  of  a  thin  slice  of 
an  altered  augite  when  viewed  under  the 
microscope.  The  black  areas  represent  purplish-black  inclusions 
in  the  almost  colorless  augite.  Note  their  arrangement  in  distinct 
lines. 

Observations  on  the  rapidity  with  which  solution  takes  place 
may  sometimes  serve  as  a  means  of  detecting  the  mode  of  crystal- 
lization of  a  substance,  when 
crystal  faces  are  not  present 
upon  it,  or  they  may  serve  to 
distinguish  between  holohe- 
dral,  hemihedral,  and  tetarto- 
hedral  forms,  or  between  the 
various  kinds  of  hemihedrism 
or  tetartohedrism;  i.e.,  they 
may  disclose  its  grade  of  sym- 
metry. 

Professors  Meyer  and  Pen- 
field  found  that  spheres  cut 
from  right-handed  and  left- 
handed  tetartohedral  crystals 
of  quartz  can  easily  be  dis- 
tinguished from  one  another  by 
subjecting  them  to  the  solvent  action  of  strong  hydrofluoric  acid. 
Etched  Figures. — A  close  inspection  of  the  faces  of  a  crystal 
that  has  been  treated  with  a  solvent  will  reveal  the  presence  on 


FIG.  305. — Result  of  treating  with  hy- 
drofluoric acid  a  sphere  of  quartz  cut 
from  right-hand  crystal.  Viewed  along 
the  vertical  axis.  There  was  practically 
no  solution  at  the  positive  ends  of  the 
lateral  axes.  (After  Pen  field.} 


SOLUTION  AND   ETCHED    FIGURES 


221 


them  of  numerous  little  hollows  and  prominences  whose  shapes 
vary  with  the  nature  of  the  substance,  the  nature  of  the  solvent, 
the  temperature  at  which  it  acts,  and  the  symmetry  of  the  face 
acted  upon.  These  are  known  as  etch  or  etched  figures.  Their 
shapes  accord  so  well  with  the  symmetry  of  the  faces  acted  upon 
that  they  have  frequently  been  used  to  determine  the  symmetry 
class  to  which  the  crystals  belong,  For  instance,  by  the  observa- 
tion of  such  figures  produced  on  cubic  crystals  of  various  sub- 
stances it  is  possible  to  distinguish  between  holohedral,  hemi- 


FIG.  306. 


FIG.  307. 


Etch  figures  on  cubic  faces  of  halite  (Fig.  306)  and  sylvite  (Fig.  307),  showing  that 
the  two  cubes  possess  different  grades  of  symmetry. 

hedral,  and  tetartohedral  cubes.  The  etch  figures  on  cubes  of 
halite  (NaCl),  for  example,  exhibit  holohedral  symmetry  (Fig. 
306),  while  those  on  cubes  of  sylvite  (KC1)  indicate  the  symmetry 
of  gyroidal  hemihedrism  (see  Fig.  307). 

All  the  etch  figures  are  bounded  by  planes  that  are  subject  to 
the  same  crystallographic  laws  as  the  planes  on  the  crystals  of 
the  etched  substance.  By  their  means  it  has  been  learned  that 
many  crystals  that  were  formerly  supposed  to  be  holohedral  are 
in  reality  hemihedral,  being  made  up  of  combinations  of  plus  and 
minus,  or  of  right  and  left  hemihedrons.  In  this  way  cubes  of 
sylvite,  as  indicated  above,  have  been  shown  to  be  gyroidally 
hemihedral  (Fig.  307),  hexagonal  pyramids  of  apatite  have  been 
shown  to  be  pyramidal-hemihedral,  and  the  pyramid  face  of 
quartz  to  be  tetartohedral. 

Muscovite  for  a  long  time  was  believed  to  be  orthorhombic. 
It  possesses,  however,  the  optical  properties  of  a  monoclinic 
mineral.  Etching  with  hydrofluoric  acid  produces  little  figures 


222 


PHYSICAL   CRYSTALLOGRAPHY 


on  the  basal  plane  somewhat  similar  in  shape  to  those  shown  in 
figure  308.  These  are  symmetrical  about  a  single  plane.  Musco- 
vite thus  is  in  reality  a  monoclinic  mineral  with  an  orthorhombic 
habit.  The  figures  produced  on  the  macropinacoid  of  calamine 


FIG.  308. — Etch  figures  on 
basal  plane  of  muscovite 
which  prove  the  mineral  to 
be  monoclinic. 


FIG.  309.— Etch  figures  on 
orthopinacoid  of  calamine 
showing  its  hemimorphic 
character. 


A  B 

FIG.  310. — Etch  figures  on  right-hand  (A)  and  left-hand  (B)  quartz  crystals, 
disclosing  their  low  grade  of  symmetry.     (After  Penfield.} 

(orthorhombic  H2Zn2SiO5)  by  dilute  hydrochloric  acid  are 
indicated  in  figure  309.  They  are  symmetrical  about  a  vertical 
line,  but  are  terminated  differently.  The  mineral  is  hemi- 
morphic. The  pyramidal  faces  of  quartz  appear  to  belong  to 


SOLUTION    AND    ETCHED    FIGURES 


223 


the  hexagonal  pyramid.  Etching  with  hydrofluoric  acid  produces 
unsymmetrical  figures  that  are  differently  arranged  on  contiguous 
planes,  but  similarly  arranged  on  alternate  ones  (see  Fig.  310). 
The  pyramid  thus  breaks  up  into  two  rhombohedrons,  and 
both  of  these  are  unsymmetrical.  The  forms  are  tetartohedral. 
Figure  311  is  a  reproduction  of  a  micro-photograph  of  the 


FIG.  311. — Photograph  of  greatly  magnified  etch  figures  on 
the  basal  plane  of  apatite.     (After  Baiimhauer .} 

etched  figures  produced  on  the  basal  plane  of  apatite  (Ca3(PO4)2 
+  (CaCl).  CaPO4)  by  dilute  sulphuric  acid.  Usually  the  figures 
are  not  as  sharply  defined  as  these,  but  they  are  nearly  always 
sufficiently  distinct  to  exhibit  clearly  the  symmetry  of  hexagonal 
pyramids  of  the  third  order. 

The  amount  of  work  done  in  the  investigation  of  etched 
figures  has  been  enormous,  and  the  results  reached  as  the  result 
of  this  study  have  had  an  immense  influence  upon  the  views 
of  mineralogists  concerning  crystallization. 


PART  III. 
CHEMICAL  CRYSTALLOGRAPHY. 


CHAPTER  XXII. 
ISOMORPHISM  AND  POLYMORPHISM. 

Chemical  Compounds. — Crystallized  substances  are  definite 
chemical  compounds  the  characteristics  of  which  are  dependent 
upon  the  nature,  the  number,  and  the  arrangement  of  the  chemical 
atoms  in  their  molecules,  and  upon  the  arrangement  of  their 
molecules.  We  usually  refer  to  the  nature  and  the  number  of 
the  atoms  in  the  molecule  as  its  composition,  while  we  designate 
the  arrangement  of  the  atoms  as  its  constitution.  Two  substances 
may  possess  the  same  composition,  but  be  differently  constituted, 
hence  may  be  different  things,  endowed  with  different  properties. 
Differences  in  the  arrangement  of  the  molecules  produce  differ- 
ences in  crystallization. 

The  difference  in  crystallization  and  in  the  physical  properties 
of  two  substances  that  are  identical  in  composition,  as,  for  instance, 
the  hexagonal  and  the  orthorhombic  forms  of  CaCO3,  may  be 
due  to  differences  in  the  constitution  of  their  molecules,  and 
this  difference  in  constitution  may  be  the  ultimate  cause  of  their 
difference  in  crystallization.  Since  the  two  substances  are  dif- 
ferent things  exhibiting  different  properties,  they  have  been  given 
different  names.  One,  the  hexagonal  form,  is  called  calcite,  while 
the  other  has  been  named  aragonite. 

Polymorphism.— When  a  substance  crystallizes  in  two  dis- 
tinct forms  it  is  said  to  be  dimorphous;  when  in  three  orms, 
trimorphous,  etc.  In  general  it  is  said  to  be  polymorphous. 

We  can  scarcely  speak  with  accuracy  of  one  substance  crys- 
tallizing in  several  forms.  If,  as  has  been  assumed,  the  morpho- 
logical properties  of  bodies  are  functions  of  their  chemical  compo- 
sition, it  must  necessarily  follow  that  bodies  with  different  proper- 
ties are  different  substances  chemically.  Two  substances  possess- 
ing exactly  the  same  number  of  the  same  elements  in  their  mole- 
cules may,  nevertheless,  be  distinct  substances  by  virtue  of  the 
difference  in  the  arrangement  of  these  elements  within  the  mole- 
cules. They  are  frequently  spoken  of  as  the  same  substance, 

227 


228  CHEMICAL    CRYSTALLOGRAPHY 

because   their   composition   may   be   represented   by   the   same 
chemical  formula  (cf.  p.  4). 

For  instance,  the  carbonate  of  calcium,  which  may  be  repre- 
sented by  the  formula  CaCO3  crystallizes  in  the  hexagonal  system 
as  calcite,  and  in  the  orthorhombic  system  as  aragonite,  as  has 
already  been  stated.  It  is  therefore  said  to  be  dimorphous.  It 
is  probable,  however,  that  one  of  these  substances  would  better 
be  represented  by  the  formula  (CaCO3)^  signifying  that  its 
molecule  contains  more  atoms  than  does  the  molecule  of  the 
other.  If  x=2,  the  formula  becomes  (CaCO3)2,  in  which  case 
the  difference  in  composition  between  the  two  substances  might 
be  represented  by  the  constitutional  formulas : 

x°v  /°\    /°\    /°\ 

Ca<      }C=O,    for    calcite,    and    Ca^      >C(      >C<       )Ca, 

\o/  \o/     \o/     \o/ 

for  aragonite. 

Polymorphous  modifications  of  a  substance  are  the  result  of 
variations  in  the  conditions  under  which  its  different  forms  are 
produced.  These  differ  in  purely  physical  properties  as  widely 
as  they  do  in  geometrical  properties.  Orthorhombic  sulphur 
crystallizes  from  solution.  Monoclinic  sulphur  is  obtained  by 
cooling  a  fused  mass.  The  one  is  transformed  into  the  other 
at  about  96°.  Orthorhombic  sulphur  melts  at  113.5°,  and 
monoclinic  sulphur  at  119.5°.  Tne  density  of  the  former  is  2 . 05 
and  of  the  latter  i .  96. 

Mercuric  iodide  is  another  substance  possessing  well-known 
modifications.  From  solution  it  crystallizes  as  red  tetragonal 
crystals;  but  from  a  fused  mass  it  separates  as  yellow  orthorhom- 
bic crystals.  The  red  variety  passes  into  the  yellow  variety  when 
it  is  heated  to  126°. 

PARTIAL    LIST    OF    POLYMORPHOUS    BODIES. 

Dimorphs. 

Pyrite  (regular)  FeS2  (orthorhombic)  Marcasite 

Arsenolite  (regular)  As2O3  (monoclinic)    Claudetite 

Calcite  (hexagonal)  CaCO3  (orthorhombic)  Aragonite 

Yellow  (orthorhombic)         HgI2  (tetragonal)  red 


ISOMORPHISM  AND    POLYMORPHISM 


229 


Trimorphs 

Ti02 
Anatase  (tetragonal,  a  :  c=i  :  .71) 

Rutile    (tetragonal,  a  :  c=i  :  .64) 
Brookite  (orthorhombic) 


Sp.Gr.  =  3.  84.  Double  Re- 

fraction —  . 
Sp.Gr.  =  4.24.  Double  Re- 

fraction +  . 
Sp.Gr.  =4.06.  Double  Re- 

fraction +  . 


Al2SiO5 

Cyanite  (triclinic)                         Sp.Gr.  =  3.60.  Hardness  5-7. 

Sillimanite  (orthorhombic)         Sp.Gr.  =  3.24.  Hardness  6-7. 

Andalusite  (orthorhombic)         Sp.Gr.  =  3.18.  Hardness  7. 


Quartz  (hexag.)       Tridymite   (orthorh.)       Asmanite     (tetrag.) 

Tetramorph. 

Sulphur  may  be  orthorhombic,  monoclinic  with  a  :  b  :  c  = 
.996  :  i  :  .999,  /3=95°46/,  monoclinic  with  a  :b  :c=i  .06  :  i  :  .71, 
,9=  91°  47',  and  hexagonal. 

Isomorphous  Compounds.  —  A  comparative  study  of  crystal- 
lized bodies  has  shown  that  those  possessing  analogous  composi- 
tions often  possess  also  the  same  general  crystalline  form. 

Crystals  are  of  the  same  general  crystalline  form  when  they 
possess  the  same  grade  of  symmetry  and  the  same  habit  and  have 
their  corresponding  interfacial  angles  of  nearly  the  same  value. 
Compounds  possess  analogous  compositions  when  their  for- 
mulas are  of  the  same  type,  as  Na2CO3  and  K2CO3  or  Na3PO4, 
Na3AsO4  and  K3AsO4. 

Analogous  compounds  may  be  regarded  as  derived  from  one 
another  or  from  some  common  source  by  the  replacement  of 
single  elements  or  groups  of  elements  by  certain  other  nearly 
allied  elements  or  groups  of  elements.  For  instance,  by  replace- 
ment of  the  hydrogen  in  H2CO3  the  following  series  of  compounds 
may  be  derived:  MgCO3,  CaCO3,  FeCO3,  MnCO3,  etc.  These 
possess  analogous  compositions. 


230  CHEMICAL    CRYSTALLOGRAPHY 

Isomorphous  bodies,  in  brief,  are  those  possessing  similar  crys- 
tal forms.  Isomorphism,  or  the  property  of  being  isomorphous, 
is  limited  to  bodies  of  analogous  compositions. 

The  following  short  list  of  isomorphous  minerals  will  illus- 
trate the  meaning  of  the  term: 

PARTIAL   LIST    OF   ISOMORPHOUS    GROUPS. 

Orthorhombic  Hexagonal  Orthorhombic 

Forsterite   Mg2SiO4  Calcite  CaCO3  Diaspore        AIO(OH) 

Tephroite  Mn2SiO4  Magnesite          MgCO3  Goethite        FeO(OH) 

Fayalite        Fe2SiO4  Spherocobaltite  CoCO3  Manganite  MnO(OH) 

Siderite  FeCO3 

Rhodochrosite  MnCO3 

Smithsonite         ZnCO3 

Morphotropism. — Although  isomorphous  compounds  possess 
the  same  general  crystallographic  habit,  there  are  no  two  of  them 
exactly  alike.  With  a  change  in  the  chemical  composition  of  any 
substance  there  is  a  corresponding,  though  sometimes  but  slight, 
change  in  its  morphological  and  physical  properties.  That  partial 
change  which  is  effected  in  the  crystallization  of  a  substance  by  the 
replacement  of  one  of  its  constituents  by  some  other  element  or  group 
of  elements  is  known  as  morphotropism.  This  usually  consists  in 
a  slight  change  in  the  axial  ratio  of  its  crystals  and  a  correspond- 
ing change  in  the  values  of  their  interfacial  angles.  The  change 
in  morphological  properties  is,  of  course,  attended  with  changes  in 
physical  properties.  The  character  and  amount  of  morphotropic 
change  produced  by  the  introduction  of  an  element,  or  group  of 
elements,  into  any  compound  is  known  as  its  morphotropic  action. 

The  variation  in  axial  ratio  shown  by  members  of  two  iso- 
morphous groups  is  illustrated  below: 

Hexa  gonal-He  mihedral  Orthorhombic 

Calcite,  CaCO3,                a:c=  10.8543  Mg2SiO4,   a  :  b  :  c=  .4666  :  i  :  .  5868 

Rhodochrosite,  MnCO3,  a  :c  =  10.8259  Mn2SiO4,  a  :  b  :  c=  .4621  :  i  :  .5914 

Siderite,  FeCO3,                 a:c=  :  0.8191  Fe2SiO4,     o  :  6  :c=  .4584  :  i  :  .5791 

Magnesite,  MgCO3,          a  :  c  =  :  0.8095 

Smithsonite,  ZnCO3,         a:c=  :  0.8062 

Isomorphous  Mixtures. — Since  isomorphous  compounds 
have  the  same  crystallographic  form,  they  naturally  tend  to  crystal- 


ISOMORPHISM  AND    POLYMORPHISM  231 

lize  together  when  present  in  the  same  crystallizing  solution.  The 
result  of  this  crystallization  is  a  mixture  of  the  different  compounds 
constituting  individual  crystals,  which  to  all  tests  appear  homo- 
geneous throughout  their  entire  masses  when  the  volume  of  the 
crystallized  substance  that  separates  is  small  as  compared  with  the 
volume  of  the  solution.  Mixtures  of  this  kind  are  called  iso- 
morphous  mixtures.  They  are  very  common  among  minerals, 
perhaps  more  common  among  the  silicates  than  are  simple 
compounds. 

Isomorphous  compounds  may  be  defined  from  this  point  of  view 
as  those  which  may  crystallize  together  or  which  may  unite  in  various 
proportions  to  produce  homogeneous  crystals.  The  proportions  of 
the  substances  in  the  mixed  crystals  bears  no  fixed  relation  to  their 
molecular  weights,  as  is  the  case  in  molecular  compounds.  On 
the  other  hand,  they  may  occur  in  practically  any  proportion, 
depending  upon  the  composition  of  the  solution  from  which  the 
crystals  separate.  In  some  cases  there  is  manifested  a  tendency 
for  the  substances  to  crystallize  together  in  certain  proportions 
rather  than  in  others,  but  in  many  other  cases  they  may  crystallize 
in  all  proportions  possible. 

The  property  of  forming  mixed  crystals  is  regarded  as  so 
characteristic  a  feature  of  isomorphous  substances  that  substances 
are  not  generally  regarded  as  isomorphous  until  they  have  been 
made  to  crystallize  together  or  have  been  found  in  crystals  in 
various  proportions  which  are  not  in  the  ratio  of  their  molecular 
weights. 

Illustration  of  Properties  of  Isomorphous  Mixtures. — An 
instructive  illustration  of  the  dependence  of  the  physical  and  mor- 
phological properties  of  substances  upon  their  chemical  composi- 
tion is  afforded  by  the  crystals  composed  of  mixtures  of  MgSO4  + 
7H2O  and  Zn  SO4+7H2O.  These  compounds  are  known  as 
epsomite  and  goslarite  when  found  in  nature.  They  crystallize  in 
the  sphenoidal  division  of  the  orthorhombic  system,  and  they  form 
mixed  crystals  containing  all  proportions  of  the  two  molecules. 
The  properties  of  the  crystals  are  determined  by  the  percentage  of 
the  magnesium  (or  the  zinc)  salt  present  in  them.  The  following 
table  exhibits  these  relations  for  a  few  of  the  mixtures  investigated : 


TOO 

i  .  6760 

89°  25' 

74-44 

1.7472 

89°  15' 

57-59 

1.7977 

89°  8' 

35-64 

i  .  8604 

89°  i' 

i8.ii 

i  .  9094 

ooo  ^  ./ 

as  54 

0 

i  .  9600 

88°  48' 

232  CHEMICAL   CRYSTALLOGRAPHY 

PROPERTIES    OF   ISOMORPHOUS   MIXTURES    OF 

MgSO4.7H2O  and  ZnSO4.7H2O. 

Per  cent,  of  Mg  Molecule      Sp.  Gr.      Angle  ooP/\  ooP         Optical  Axial  Angle 

78°  18'    o" 
76°  42'  47"* 

74°    3' 45"* 

70°  53' 

In  general  the  physical  properties  of  crystals  composed  of 
mixtures  of  isomorphous  compounds  are  functions  of  the  com- 
pounds crystallizing  together,  being  intermediate  between  the 
two  pure  compounds.  This  relationship  between  the  physical 
properties  and  the  chemical  composition  of  mixed  crystals  is  so 
close  that  Retgers  declares  that  "  Two  substances  are  truly  isomor- 
phous only  when  the  physical  properties  of  their  mixed  crystals  are 
continuous  functions  of  their  chemical  composition" 

Formulas  of  Isomorphous  Mixtures. — Although  isomor- 
phous mixtures  are  not  definite  combinations  of  elements  in  the 
same  sense  as  are  simple  compounds,  nevertheless,  they  exist 
so  frequently  that  some  means  must  be  decided  upon  for  the 
representation  of  their  composition.  The  logical  method  of 
writing  their  formulas  is  to  designate  the  number  of  molecules  of 
each  of  the  substances  entering  into  the  mixture.  This  method, 
however,  would  require  a  different  formula  for  every  different 
mixture  possible.  Since  the  number  of  these  is  often  large,  a  great 
number  of  formulas  would  be  demanded,  and  some  of  them  would 
be  very  complicated. 

In  practice  it  is  deemed  sufficient  to  indicate  by  the  formula  the 
nature  of  the  molecules  in  the  mixture  without  specifying  exactly 
their  proportions.  This  is  done  by  enclosing  the  symbols  of  the 
mutually  replaceable  elements  or  groups  of  elements  in  parentheses 
with  a  comma  between  them,  and  writing  the  symbols  of  the 
remaining  elements  in  the  usual  manner.  Such  a  formula,  as, 
for  instance,  (Fe,Mg)2SiO4,  indicates  that  iron  and  magnesium 

*Approximate. 


ISOMORPHISM  AND    POLYMORPHISM  233 

silicates  are  present  in  different  proportions  in  a  series  of  com- 
pounds possessing  the  same  general  crystallographic  and  physical 
features. 

ILLUSTRATIONS    OF    FORMULAS    OF   ISOMORPHOUS    MIXTURES. 

Spinel  is  MgAl2O4.  Iron  spinel  is  MgFe2O4.  An  isomor- 
phous  mixture  of  these  is  represented  by  Mg(Fe,Al)2O4. 

Barite=BaSO4.  Celestite  =  SrSO4.  Barito-celestite  =  (Ba,Sr) 
SO4.  Tetrahedrite=(Ag2,Cu2,Fe,Zn,Hg)4)(Sb,As)2S7. 

Isodimorphous  Groups. — A  series  of  isomorphous  poly- 
morphs  is  an  isopolymorphous  group.  Series  of  isomorphous 
compounds  each  one  of  which  is  a  dimorph  are  very  common. 
Such  groups  are  known  as  isodimorphous  groups.  A  simple 
example  is  the  following: 

Regular.  Orthorhombic. 

Arsenolite,  As2O3  Claudetite 

Senarmonite,  Sb2O3  Valentinite 

Double  Salts. — Many  crystallized  substances  that  are  not 
isomorphous  mixtures  appear  to  consist  of  a  combination  of  mole- 
cules which,  however,  unite  in  definite  proportions,  and  not  in 
many  different  proportions  like  the  molecules  in  isomorphous 
mixtures.  They  are  known  as  double  salts  because  they  appear 
to  be  made  up  of  portions  that  may  exist  independently  as  simple 
compounds.  An  illustration  in  case  is  sodium  silver  chloride, 
whose  formula  is  NaCl  +AgCl,  or  cryolite,  a  monoclinic  mineral 
with  the  composition  Na3AlF6  or  3NaF.AlF3.  Substances  con- 
taining water  of  crystallization  also  belong  to  this  class  of  mo- 
lecular compounds.  Gypsum,  CaSO4  +2H2O,  is  an  example. 

It  is  not  at  all  certain  that  the  double  salts  differ  in  any  essen- 
tial respect  from  ordinary  atomic  molecules.  They  are  referred 
to  here  only  for  the  purpose  of  emphasizing  the  fact  that  isomor- 
phous compounds  form  mixed  crystals  in  which  the  proportions 
of  the  components  present  may  vary  with  varying  conditions  dur- 
ing growth,  whereas  the  crystals  of  a  double  salt  the  components 
of  which  are  not  members  of  an  isomorphous  series  have  a  defi- 
nite composition  which  is  invariable. 


INDEX. 


Absorption  of  light,  199-202 
Acicular,  134 
Aggregates,  143^147 

botryoidal,  146-147 
cryptocrystalline,  144 
crystalline,  143,  144-147 
fibrous,  146 
globular,  145 
granular,  145 
lamellar,  145 
radial,  145 
sheaf-like,  147 
Allochromatic,  197,  200 
Amorphous  substances,  6,  160,  177,  180, 

214 

Analogue  pole,  216 
Antilogue  pole,  216 
Axes,  crystallographic,  17 

of  hexagonal  system,  55-56 
of  isometric  system,  32-33 
of  monoclinic  system,  116 
of  orthorhombic,  102-103 
of  regular  system,  32-33 
of  tetragonal  system,  88 
of  triclinic  system,  125 
projection  of,  169-173 
Axial  angle,  207-208 

apparent,  208 
true,  207 

ratio,    determination    of,    in    hex- 
agonal system,  57-60 
determination   of,    in   tetragonal 

system,  89-90 
of  hexagonal  system,  56-57 
of  monoclinic  system,  116-117 
of  orthorhombic  system,  103—104 
of  tetragonal  system,  89-90 
of  triclinic  system,  1 26 
Axis,  twinning,  152 


B 

Basal  pinacoid,  64,  119 

Biaxial  crystals,  206 

Boiling  point,  211 

Botryoidal,  146,  147 

Brachyaxis,  103,  126 
domes,  108,  128 
hemi  domes,  128 
hemiprisms,  127-128 
pinacoid,  109,  128-129 
prisms,  107-108,  127-128 
pyramid,  107-108,  126-127 
series,  107-108 
tetrapyramids,  127 


Chemical  compounds,  227 

Classes  of  symmetry: 

dihexagonal  bipyramidal,  60-69 
ditetragonal  bipyramidal,  87-95 
ditrigonal  bipyramidal,  78-80 
ditrigonal  pyramidal,  80-81 
domatic,  121-122 
dyakisdodecahedral,  46-48 
hexagonal  bipyramidal,  75-76 
hexagonal  trapezohedral,  76-77 
hexoctahedral,  31-38 
hextetrahedral,  48-51 
orthorhombic  bipyramidal,  105—110 
orthorhombic  bisphenoidal,  112-114 
orthorhombic  pyramidal,  110-112 
pentagonal  icositetrahedral,  46 
pinacoidal,  125-131 
prismatic,  117-121 
sphenoidal,  122-124 
tetragonal  bipyramidal,  98-99 
tetragonal  scalenohedral,  95-98 
tetragonal  trapezohedral,  100 
trigonal  scalenohedral,  70-74 
trigonal  trapezohedral,  82-86 


235 


236 


INDEX. 


Cleavage,  181-183 

planes  of,  181-182 

symmetry  of,  182-183 
Clinoaxis,  116 

domes,  119 

hemipyramids,  118-119 

pinacoids,  119-120 

prisms,  119 
Closed  forms,  68,  109 
Coefficient  of  elasticity,  180 

of  expansion,  210-211 
Cohesion,  181 
Colloids,  6,  8,  1 60 
Color,  196-197 

allochromatic,  197,200 

ideochromatic,  197,200 
Columnar,  134 
Combinations,  43-44 

hexagonal,    68-69,    73-74,    80-8 1, 
85-86 

isometric,  38-39,  51-53 

monoclinic,  120-121 

orthorhrombic,  108,  109-110 

tetragonal,  94-95,  99-100 

triclinic,  129-130 
Composition  face,  152 
Conchoidal  fracture,  187-188 
Conduction  of  electricity,  215 

of  heat,  209-210 
Congruent  forms,  47 
Constancy  of  interfacial  angles,  14-16 
Contact  twins,  152-153 
Contraction  in  crystals,  210-211 
Corrosion  of  planes,  136 
Cryptocrystalline,  144 
Crystal  angle,  10 

axes,  17 

biaxial,  206 

definition  of,  8 

drawing,  168-174 

edge,  10 

form,  21,  27 

group,  143 

inclusions,  141 

individual,  8 

liquid,  8 

particles,  4-5 

projection,  164-174 


Crystal,  structure  of,  5-7 

uniaxial,  206 

Crystalline  bodies,  6-9,  177 
Crystallites,  141 
Crystallization,  8-9 
Crystallographic  axes,  17 

constants    of    monoclinic    system, 

116-117 

of  triclinic  system,  1 26 
Crystallography,  9 

definition  of,  xi 

laws  of,  9-10 

systems  of,  28,  29-30  • 

comparison  of,  130-131 
Cube,  37 

Curvature  of  planes,  136 
Cyclic  twins,  155 


Dana's  notation,  22 
Deformed  crystals,  134-135 
Dendritic  growth,  149 
Density.     See  specific  gravity. 
Determination   of    axial    ratio    in   tet- 
ragonal system,  89-90 

in  hexagonal  system,  57-60 
Diamagnetic,  214 
Diathermous,  209 
Dichroiscope,  201 
Dichroism,  201 
Dihexagonal  bipyramid,  61-62 

prism,  64 

series,  62-64 
Dimorph,  162 

Dimorphous  substances,  227-229 
Diploid,  47 

Distorted  crystals,  132-133 
Ditetragonal  bipyramid,  90-91 

series,  91-92 
Ditrigonal  bipyramids,  79-80 

prisms,  79-80 

of  second  order,  84 
Dodecahedron,  36 
Double  refraction,  204-206 

explanation  of,  205-206 
Double  salts,  233 
Drawing  of  crystals,  168-174 


INDEX. 


237 


Druse,  149-150 
Dyakisdodecahedron,  47 


Elasticity,  180 

Electrical  properties,  214-218 
Electricity,  conduction  of,  215 
Enantiomorphous  forms,  83 
Etched  figures,  136,  220-223 

symmetry  of,  221-223 
Expansion  of  crystals,  210-211 
Extraordinary  ray,  205 


Fibrous  aggregates,  146 
Fluid  inclusions,  139,  142 
Fluorescence,  202 
Fossilization,  163 
Fourlings,  154,  157 
Fracture,  186-187 
Fusibility,  scale  of,  212-213 
Fusing  points,  211 

table  of,  212 
Fusion,  2ii— 212 


Gas  inclusions,  139-142 
Glass  inclusions,  141 
Gliding  planes,  183-184 
Globular  aggregates,  145 
Goniometer,  11-13 
Granular  aggregates,  145 
Groundform,  33 
Gyroidal  hemihedrism,  46 

H 

Habit  of  crystals,  14,  133-134 
acicular,  134 
columnar,  134 
prismatic,  134 
tabular,  134 

Hardness,  187-192 

curves  of,  190-191 
determination  of,  189 
differences  in,  189,  190 
list  of  relative,  188 
scale  of,  187 

Hauy,  x 


Heat,  conduction,  209-210 

transmission,  209 
Heavy  solutions,  194 

Klein,  194 

Retgers,  194 

Rohrbach,  194 

Thoulet,  194 
Hemibrachydomes,  128 
Hemiclinodomes,  122 
Hemihedrism,  31,  342 

in  hexagonal  system,  69-82 

in  isometric  system,  41-53 

in  monoclinic  system,  123—124 

in  orthorhombic  system,  112-114 

in  tetragonal  system,  95-101 

law  of,  42-43 
Hemimacrodomes,  128 
Hemimorphism,  31,  41-42 

in  hexagonal  system,  80-8 1 

in  monoclinic  system,  121-123 

in  orthorhombic  system,  110-112 

in  tetragonal  system,  101 
Hemiprisms,  122,  127-128 
Hemipyramids,  118 
Hexagonal  prisms  of  first  order,  65-66 
of  second  order,  66-67 

pyramids  of  first  order,  65-66 
of  second  order,  66-67 

rhombohedrons,  71-73 

scalenohedrons,  71 
Hexagonal  system,  54—86 

axes  of,  55-56 

axial  ratio  in,  56-57 

determination  of,  57-60 

combinations  in,  68-69,  73-74 

comparison  with  tetragonal  system, 
54-55,  87 

general  form  in,  61 

groundform  in,  56 

hemihedral  division,  69-82 

hemimorphism  in,  80— 8 1 

holohedral  division,  60-69 

symmetry  of,  54-55 

tetartohedrism  in,  82-86 

three  series  of  holohedrons  in,  64-65 

trapezohedrons,  77 
Hexoctahedron,  34 
Hextetrahedron,  49 


238 


INDEX. 


Holohedron,  31 

Hopper- shaped  crystals,  137 


K 


Klein  solution. 


194 


Icositetrahedron,  35 
Ideal  forms,  132 
Ideochromatic,  197,  200 
Idiomorphic  form,  3,  4 
Imperfections  in  crystals,   132-142 
Inclusions,  138-142 

crystal,  141 

fluid,  139,  142 

gas,  139,  141,  142 

glass,  140 

Index  of  refraction,  203-204 
Indices,  23 
Intercept    on    lateral    axes,    hexagonal 

system,  58-60 
Interfacial  angles,  10,  n 
constancy  of,  14-16 
measurement  of,  11-13 

edge,  10 
Interpenetration   twins,    153,   154,   156, 

157,  158 

Irregularities  in  crystals,  132 

Isodimorphous  group,  233 

Isometric  system,  31-53 
axes  of,  32-33 

combinations  in,  38-39,  43-44 
derivation  of  forms  in,  34 
determination  of  forms  in,  37-38 
general  form  in,  33-34 
groundform  in,  33 
hemihedral  division,  41-53 
holohedral  division,  31-40 
symmetry  of  31-32 
tetartohedrism  in,  53 

Isomorphism,  227,  229-230 

Isomorphous  compounds,  230—231 
formulas  of,  232-233 
properties  of,  231-232 
groups,  230 
mixtures,  230-231 

Isotherms,  210 


J 


Jolly  balance,  192,  193 


Lamellar  aggregates,  145 

Law  of  constancy  of  interfacial  angles, 

14-16 

of  hemihedrism,  42—43 
of  rationality  of  the  indices,  17 
of  simple  mathematical  ratios,  17-20 

application  of,  22 
of  symmetry,  28 
of  tetartohedrism,  42-43 

Light,  absorption  of,  199 
reflection  of,  196 
refraction  of,  203-208 
transmission  of,  199 

Liquid  crystals,  8 

List  of  fusing  points,  212 

of  hexagonal  hemihedrons,  81 

of  isometric  hemihedrons,  51 

of  isodimorphous  groups,  230 

of  isomorphous  groups,  230 

of  polymorphous  bodies,  228-229 

of  specific  gravities,   195 

of  tetragonal  hemihedrons,  101 

Luster,  198,  199 

M 
Macroaxis,  103,  126 

domes,  108—109,  128 

pinacoid,  109,  128    129 

prism,  107,  127-128 

pyramids,  107,  126-127 

series,  106 

tetrapyramids,  127 

hemiprisms,  127-128 

hemidomes,  1 28 

Magnetic  properties  of  crystals,  214 
Magnetism,  214 
Mechanical  properties  of  crystals,  180- 

*95 

Merohedrism,  41 

Microlites,  141 

Miller's  system  of  notation,  23—24 

Mimicry,  158 

Mineral,   definition  of,  ix 


INDEX. 


239 


Mineralogy,  history  of,  x 
purpose  of,  ix 
subdivisions  of,  xi-xii 
Mohs's  scale  of  hardness,  187 
Molecules,  4-6,  7-8 
Monoclinic  domes,  19 
hemiclinodomes,  122 
hemipyramids,  118-119 
orthodomes,  119 
pinacoids,  119-  120 
prisms,  119 
system,  115-124 
axes  of,  116 

combinations  in,  120-121 
crystallographic      constants     in, 

116-117 

groundforms  in,  116—117 
hemihedral  division,  123-124 
hemimorphism  in,  121-123 
holohedral  division,  117-121 
symmetry  of,  115-116 
tetraorthodomes,  122 
tetrapyramids,  122 
Morphological  mineralogy,  9 
Morphotropism,  16,  230 

N 
Naumann's  system  of  notation,  21-22, 

24 

Negative  crystals,  139 

Notation,  crystallographic,  20 
Miller's  system  of,  23-24 
Naumann's  system  of,  21-22,  24 
Weir's  system  of,  20-21 


Octahedron,  33,  37 
Open  forms,  68 
-Optical  axes,  207 

properties  of  crystals,  196-208 
Ordinary  ray,  205 

Orthorhombic  brachydomes,  108-109 
pinacoids,  108-109 
prisms,  107-108 
pyramids,  107-108 
macrodomes,  108-109 
pinacoids,  108-109 
prisms,  106-107 


Orthorhombic  pyramids,  106-107 
system,  102-114 

axes  of,  102-104 

closed  forms  in,  109 

combinations  in,  108,  109-110 

general  form  in,  105 

groundform  in,  103 

hemihedral  division,  112-114 

hemimorphism  in,  iio-ni 

holohedral  division,  105-112 

symmetry  of,  102 

three  series  of  forms  in,  105 
unit  prisms,  105-106 

prisms,  105-106 


Parallel  growths,  143,  147-151 
Paramagnetic  substance,  214 
Parameters,  18-19 
Paramo  rphs,  162 
Partial  forms,  41 
Parting,  186 

Penetration  twins,  153,  154 
Pentagonal  dodecahedrons,  47-48 
hemihedrons,  46-48 
icositetrahedron,  46 
Percussion  figures,  185,  186 
Phanerocrystalline,  144 
Phosphorescence,  202-203 
Physical  agencies,  178-179 
Physical  symmetry,  177,  178 
Plane  of  symmetry,  28 
Pleochroism,  200-202 
Polymorphism,  227-229 
Polymorphous  bodies,  227,  229 
Polysynthetic  twins,  155,  156 
Pressure  figures,  186 
Projection,  crystal,  164-174 
linear,  164-167,  168 
of  axes,  169-173 
isometric,  170 
tetragonal,  170 
Orthorhombic,  170-171 
monoclinic,  171 
triclinic,  171-172 
hexagonal,  172-173 
spherical,  168,  167 


240 


INDEX. 


Pseudomorphism,  explanation  of,  161 
Pseudomorphs,  160-163 
chemical,  162 
mechanical,  162-163 
Pyramidal  hemihedrism,  70,  75-76,  95, 

98-99 
Pyroelectrical  properties  of  crystals,  216- 

218 

Pyroelectricity,  216 
Pyritohedron,  47-48 

R 

Radial  aggregates,  145 
Re-entrant  angle,  152 
Reflection  of  light,  196 
Refraction,  double,  204-206 
single,  204 
of  light,  203-208 
index  of,  203-204 

Relation  between  hexagonal  pyramids 
and  prisms  of  various  orders, 
67-68,  76 
tetragonal    pyramids  and  prisms 

of  various  orders,  94 
Repeated  twins,  154-158 
Retgers  solution,  194 
Rhombohedral    hemihedrism,    70-74 
Rhombohedrons,  71-72 
short  symbols  of,  73 
Rock,  definition  of,  ix 
Rohrbach  solution,  194 


Scale  of  fusibility,  212-213 

hardness,  187 
Scalenohedrons,  71,  96-97 
Secondary  twinning,  184 
Sheaf-like  aggregates,  147 
Short    symbols    of    rhombohedra    and 

scalenohedra,  73 
Simple  mathematical  ratios,  17 
Simple  refraction,  204 
Sklerometer,  189 
Solution,  219-223 

planes  of,  219-220 

rates  of,  219 

symmetry  of,  220 


Specific  gravity,  192-195 

determination  of,  192-195 

list  of,  195 

use  of,  195 

Sphenoidal  hemihedrism,  95-98, 112-114 
Sphenoids,  97-98,  113-114 
Stalactites,  3-4,  147 
Streak,  197 

Striations,  135  -136,  137-138 
Structure  of  crystals,  5-7 
Sub-individuals,  149 
Supplementary  twins,  153-154 
Symbols  of  planes,  18-24 
Symmetry,  25-30 

axes  of,  25 

classes  of,  29 

grades  of,  28 

in  crystallography,  26-28 

law  of,  28 

of  crystal  faces,  178 

of  etched  figures,  221-223 

of  pyroelectrical  properties,  216-218 

of  solution,  220 

physical,  177-178 

planes  of,  28 

with  respect  to  lines,  25-26 
to  planes,  25 
to  points,  26 
Systems,  crystallographic,  28,  29-30 


Table  of  relative  hardness,  188 
of  fusing  points,  212 
of  hexagonal  hemihedrons,  81 
of  isometric  hemihedrons,  51 
of  tetragonal  hemihedrons,  101 

Tenacity,  180-181 

Tetartohedrism,  31,  42 

in  hexagonal  system,  82-86 
in  isometric  system,  42,  53 
in  tetragonal  system,  101 
law  of,  42  -43 

Tetrabrachypyramids,  126-127 

Tetragonal   bipyramids   of   first   order, 

92-93 

of  second  order,  93-94 
of  third  order,  98-99 


INDEX. 


241 


Tetragonal  prisms  of  first  order,  92-93 
of  second  order,  93-94 
of  third  order,  98-99 
scalenohedrous,  96-97 
series  of  various  orders,  92 
sphenoids,  97-98 
system,  87-101 
axes  of,  102-104 
axial  ratio  in,  89-90 

determination  of,  89-90 
comparison  with  hexagonal  sys- 
tem, 54-55.  87 
groundform  in,  88-90 
hemihedral  division,  95-101 
hemimorphism  in,  101 
holohedral  division,  87-95 
relations  between  forms  of  differ- 
ent orders  in,  94 
tetartohedrism  of,  101 
symmetry  of,  87-88 
Tetragonal  trapezohedrons,  100 

tristetrahedrons,  50 
Tetrahedral  hemihedrism,  48-51 
Tetrahedrons,  50-51 
Tetrahexahedrons,  36 
Tetramacropyramids,  126-127 
Tetramorph,  229 
Tetra  unit  pyramids,  126-127 
Thermal  properties  of  crystals,  209-213 
Thermoelectrical  properties  of  crystals 

215-216 

Thoulet  solution,  194 
Transfer     of     Miller     and     Maumann 

symbols,  23-24 
Trapezohedral  hemihedrism,  70,  76-77, 

95>  I0° 

tetartohedrism,  80-81 
Trapezohedrons,  83,  100 
Trichroism,  201 
Triclinic  hemidomes,  127-128 
hemiprisms,  127-128 
pinacoids,  128-129 
system,  125-130 
axes  of,  125 
combinations  in,  129-130 


Triolinic  system,  crystallographic  con- 
stants in,  1 26 
groundform  in,  126 
symmetry  of,  102 

tetrapy ramids,  1 26   1 27 
Trigonal  bipy ramids  of  first  order,  78-79 
of  second  order,  84-85 

hemihedrism,  78-80 

prisms  of  first  order,  78-89 
of  second  order,  84-85 

trapezohedrons,  83 

tristetrahedrons,  49,  50 
Trillings,  154,  156,  i57>  J58 
Trimorphs,  227,  229 
Trisoctahedron,  36 
Twinned  crystals,  143 

lamellae,  155 
Twinning,   artificial,  184 

axis,  152 

plane,  152 

secondary,  184 
Twins,  contact,  152-153 

cyclic,  155 

interpenetration,  153,  154,  156 

penetration,  153,  154 

polysynthetic,  155,  156 

repeated,  154-158 

secondary,  184 

supplementary,  153^154 

U 
Uniaxial  crystals,  206 

V 

Vaporization,  211-212 
Von  Kobell,  212-213 

W 

Weiss,  x,  21-22 
Weiss'  system  of  notation,  21 
Werner,  x 


Zonal  axis,  10 

growths,  150,  200 
Zone,  10 


.10 


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